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The  Repayment  of  Local 
and  Other  Loans 

Sinking   Funds 


The  Repayment  of  Local 
and  Other  Loans 


Sinking  Pounds 


BY 


EDWARD    HARTLEY    BURNER,    A.C.A. 

Lecturer  on  Municipal  Accounts  at  the  Manchester  Municipal 
School  of  Commerce 


THE  RONALD  PRESS  COMPANY 

198  Broadway,  New  York 
1913 


<h 


"^^^t 


PREFACE. 

In  writing  this  book  my  primary  object  has  been  to  render  it 
practically  useful  to  those  engaged  in  connection  with  the  loan 
debt  of  public  authorities  and  privately  owned  undertakings. 
It  is  the  result  of  actual  experience  extending  over  many  years, 
but  the  problems  which  have  arisen  have  been  isolated  examples 
and  have  not  occurred  in  any  regular  sequence,  but  often  at 
long  intervals.     For  this  reason  each  problem  has  been  treated 
independently  in  order  that  any  particular  adjustment  may  be 
made  by  reference  to  a  similar  example  fully  worked  out  and 
described.     This  entails  a  certain  amount  of  repetition  in  order 
that  each  chapter  may  contain  a  brief  summary,  and  a  reference 
to  the  results,   of  previous  and  subsequent  investigations.     In 
order  to  carry  this  out  in  its  entirety,  each  adjustment  has  been 
reduced  to  a  series  of  stages  briefly  stated,  and  is  accompanied 
by    detailed    statements    of    the   method    adopted,    and    actual 
calculations    upon    standard    forms    which    I    have    specially 
prepared.     But  I  have  gone  further  than  this,  and  have  grouped 
the  problems  under  the  heads  of  the  various  factors  governing 
a  sinking  fund,  namely,  the  amount  in  the  fund,  the  period  of 
repayment,  the  rate  of  accumulation  and  the  rate  of  income  to 
be    received    upon    the    present    investments.      This    orderly 
arrangement  of   isolated   practical  examples   has   compelled   a 
theoretical     method    of    treatment    which    has    been    adopted 
throughout.     In  my  own  earlier  experience  I  very  soon  realised 
that  even  after  making  the  fullest  use  of  the  published  tables 
of  compound  interest,  the  ordinary  methods  of  arithmetic  were 
utterly  inadequate  and  that  all  calculations  must  be  made  by 
logarithms;  consequently  a  full  knowledge  of  the  use  of  a  log. 
table  has  been  assumed.     The  next  practical  difficulty  which 
arose    was    that   the    ordinary    published   tables    of    compound 
interest  very  often  did  not  contain  the  required  rate  per  cent., 
and  it  therefore  became  necessary  to  make  the  calculation  by 
other  means.     For  this  reason  I  have  included  in  the  earlier 
chapters    a    brief    summary    of    the    mathematical    principles, 
showing  the  derivation  of  the  formulae  upon  which  all  the  tables 
are  constructed.     In  order,  however,  to  render  this  method  by 
formula  generally  available  I  have  reduced  all  such  calcula- 
tions to   simple   rules,    and   in    Chapter    X,    dealing  with    the 


290048 


vi  PREFACE 

various  staudard  calculation  forms,  minute  instructions  are 
given  for  finding  tlie  actual  values  of  the  whole  of  the  factors 
relating  to  any  rate  per  cent.  I  have,  in  fact,  endeavoured  to 
state  the  methods  in  such  a  manner  that  a  knowledge  of  the 
meaning  of  the  formula  is  not  required. 

Throughout  the  book  the  final  results  are  expressed  in 
decimal  form.  This  course  has  been  adopted  partly  in  order 
to  save  time,  but  primarily  to  make  the  book  applicable  to  any 
currency  of  a  decimal  nature;  and  it  is  recommended  that  all 
pro  forma  accounts  should  be  prepared  in  this  form. 

The  methods  of  adjustment  described  as  the  deductive 
method,  the  annual  increment  (ratio)  method,  and  the  annual 
increment  (balance  of  loan)  metliod,  are  not  new.  The  terms 
have  merely  been  used  for  convenience  of  reference. 

The  actual  compilation  of  the  book  has  occupied  considerable 
time,  and  it  has  not  been  written  in  the  consecutive  order  in 
which  it  is  now  presented  to  the  reader.  It  has  involved  the 
preparation  of  many  standard  forms  and  statements,  and  very 
many  calculations  Avhich  are  not  given  in  the  text.  Every  effort 
has  been  made  to  ensure  absolute  accuracy  of  detail,  but  in  a 
few  cases  the  final  decimal  figure  in  the  result  obtained  may 
not  agree  with  the  result  found  by  another  method,  and  the 
same  applies  to  the  final  figure  of  some  of  the  logs.  These 
small  differences  are  not,  however,  of  any  practical  importance. 
The  cross-references  to  other  results  have  been  carefully 
compared,  but  they  are  very  numerous  and  a  few  errors  may 
be  found. 

The  methods  adopted,  and  the  results  obtained,  have  been 
in  all  cases  verified  by  an  alternative  method  of  proof,  even 
where  it  is  not  shown  in  detail  in  the  text.  The  summaries  of 
methods,  and  the  rules  and  formulse  given  in  italics  at  the  head 
of  the  various  chapters  have  been  in  all  cases  caref  Tilly  compared 
with  the  individual  statements  and  with  each  other,  and  a 
uniform  wording  has,  as  far  as  possible,  been  adopted 
throughout. 

As  stated  in  the  Introduction,  no  attempt  has  been  made  to 
include  anything  in  the  nature  of  a  full  statement  of  the 
statutory  obligations  as  to  tlie  repayment  of  the  loan  debt  of 
local  authorities  or  the  policy  of  Parliament  with  relation 
thereto,  except  so  far  as  they  affect  the  actual  method  of 
repayment. 

In  the  final  chapters  dealing  with  the  life  of  tlie  asset,  the 
equation  of  the  period  of  repayment  and  the  incidence  of 
taxation   a   serious   attempt  has   been  made   to   elucidate   this 


PREFACE  vii 

difficult  subject.  It  has  been  treated  niaiuly  from  the  mathe- 
matical standpoint,  both  as  regards  the  annual  instalment  and 
interest  upon  the  loan.  It  is  a  matter  about  which  there  is 
naturally  some  divergence  of  opinion  as  to  its  practical 
application,  but  I  hope  that  the  considerations  here  set  forth 
will  be  of  assistance  in  arriving  at  a  definite  solution. 

The  object  of  the  book  is  entirely  practical.  I  hope  it  will 
materially  assist  those  who  are  approaching  the  consideration  of 
such  problems  for  the  first  time,  and  that  it  will  also  be  useful 
to  those  who  are  already  fully  acquainted  with  the  subject,  if 
only  because  it  attempts  to  deal  with  it  in  a  consecutive  and 
orderly  manner. 

In  conclusion,  I  wish  to  tender  my  thanks  to  my  friend  and 
colleague,  Mr.  Arthur  Holme,  A.C.A.,  who  has  assisted  me  in 
the  revision  of  the  manuscript  and  the  correction  of  the  proofs, 
and  also  to  my  son,  Edward  Gordon  Turner,  who  has  rendered 
me  very  valuable  aid  in  the  preparation  of  the  final  manuscript 
and  the  verification  of  the  various  calculations  in  the  text. 

E.H.  T. 

Manchester, 

November,  191'J. 


TO  AMEEICAN  READERS. 

In  the  early  part  of  the  year  1906,  I  was  appointed  the 
British  Accounting  Expert  to  the  National  Civic  Federation  of 
New  York  in  the  Enquiry  into  the  Municipal  and  Private 
Ownership  of  Public  Utilities  (Municipal  Trading)  in  Great 
Britain. 

The  enquiry  was  made  by  a  Commission  of  Twenty-one, 
under  the  Chairmanship  of  Melville  E.  Ingalls,  Chairman  of 
the  Board  of  Directors,  Big  Four  Railroad,  Cincinnati.  The 
details  of  the  enquiry  were  under  the  supervision  of  Dr.  Milo 
R.  Maltbie,  now  a  Member  of  the  Public  Service  Commission  of 
the  State  of  New  York.  The  Report  of  the  Commission  is 
contained  in  three  volumes,  issued  in  New  York  in  1907. 

The  enquiry  in  Great  Britain  was  confined  to  the  results  of 
municipal  and  private  ownership  and  operation  of  gas  works, 
street  railways  or  tramways,  and  electric  lighting  and  power 
undertakings.  In  the  United  States  the  enquiry  was  extended 
to  water  works. 

The  Commission  examined  24  public  and  private  under- 
takings in  London,  Birmingham,  Dublin,  Glasgow,  Leicester, 
Liverpool,  Manchester,  Newcastle-on-Tyne,  Norwich  and 
Sheffield. 

The  following  is  a  full  list  of  the  Committee  on  Investiga- 
tion :  — 

*Melville  E.  Ingalls,  Chairman  (Chairman  Board  of  Directors, 
Big  Four  Railroad),  Cincinnati. 
Albert  Shaw,  V ice-Chairman  (Editor  "  RevicAv  of  Reviews  "), 

New  York  City. 
Talcott  William  (Editorial  Writer,  the  Press),  Philadelphia. 
W.    D.    Mahon    (President  Association    Street    Railway    Em- 
ployes), Detroit. 
^Professor  Frank  J.  Goodnow  (Columbia  University),  New  York 

City. 
*Walton  Clark  (Third  Vice-President  The  United  Gas  Improve- 
ment Company),  Philadelphia. 
^Professor  Edward  W.  Bemis  (Superintendent  Water  Works), 

Cleveland. 
*Professor  John  H.  Gray  (University  of  Minnesota),  Minneapolis. 
Walter  L.  Fisher  (Special  Traction  Counsel  for  City  of  Chicago 
and  ex-President  Municipal  Voters'  League),  Chicago. 


X  PREFACE 

*Timotliy      Healey      (President      International      Brotberliood 

Stationary  Firemen),  JN^ew  York  City. 
*William   J.    Clark    (General    Manager   Foreign    Department, 

General  Electric  Company),  New  York  City. 
H.   B.    F.    Macfarland    (President   Board    of    Commissioners, 

District  of  Columbia),  Washington. 
Daniel    J.    Keefe    (President    International    Longshoremen's 

Association),  Detroit. 
*Professor  Frank  Parsons  (President  National  Public  Ownership 

League),  Boston. 
*Professor  John  R.  Commons  (Wisconsin  University),  Madison, 

Wisconsin. 
*J.  W.   Sullivan  (Editor  "  Clothing  Trades'  Bulletin  "),   New 

York  City. 
*F.    J.     McNulty    (President    International    Brotherhood    of 

Electrical  Workers),  Washington. 
*Albert  E.  Winchester  (General  Superintendent  City  of  South 

Norwalk  Electric  Works),  South  Norwalk,  Conn. 
*Charles    L.     Edgar     (President     The     Edison     Electric     and 

Illuminating  Company),  Boston. 
*Dr.  Milo  R.  Maltbie  (member  of  the  Public  Service  Commis- 
sion),   New   York  City;    now   a   member   of   the   Public 

Service  Commission  of  the  State  of  New  York. 
Leo  S.  Rowe  (University  of  Pennsylvania),  Philadelphia. 
*Edward  A.  Moffet,  Secretary  (Editor  "Bricklayer  and  Mason"), 

Indianapolis,  Ind. 

The  investigation  in  Great  Britain  was  made  by  the  members 
of  the   Committee   indicated  by   an   asterisk. 

The    following   summary    of    the    financial    details    of    the 
undertakings  examined  will  give  some  idea  of  the  magnitude  of 
the  enquiry. 
Loan  Capital :  — 
Outstanding  : 

Municipal       £18,8GG,648 

Private    6,144,945 

£25,011,593 


Repaid  by  means  of  Sinking  Fund  : 

Municipal        3,868,301 


£28,879,894 
Share  Capital  or  Stocly,  Private    12,146,504 


Total  Capital  raised     £41,026,398 


PREFACE  xi 


Capital  expended  on  WorJcs  :  — 

Municipal        £24,517,370 

Private    18,266,368 


Gross  Revenue  for  the  year  under  Examination:- 

Municipal        £6,091,821 

Private    3,956,936 


£42,783,738 


£10,048,757 


The  examination  by  the  experts  was  commenced  early  in 
March,  1906,  and  continued  without  interruption  until  the  end 
of  July,  1906.  The  members  of  the  Commission  were  engaged 
in  Great  Britain  from  30th  May  to  the  end  of  -July,  1906. 

The  expert  assistance  was  divided  between  the  Technical 
and  Accounting  sides.  In  each  class  of  undertaking  an 
American  and  a  British  technical  expert  made  a  joint  investiga- 
tion, valuation,    and   report. 

The  British  technical  experts  were  :  — 
Gas  Works,  Mr.  William  Newbigging,  Consulting  Engineer,  of 

Manchester. 
Tramways  (Street  Railroads),  Mr.  J.  H.  Woodward,  Consulting 

Engineer,  London. 

The  American  technical  experts  were  :  — 
Electricity  Supply,  Mr.  Albert  E.  Winchester,  General  Supt. 

City  of  South  Norwalk  Electric  Works,  South  Norwalk, 

Conn. 
Gas  Works,   Mr.   J.  B.   Klumpp,   Inspecting  Engineer  of  the 

United  Gas  Improvement  Company,  Philadelphia. 
Tramways   (Street   Railroads),    Mr.   Norman    McD.    Crawford, 

Consulting  Engineer,  Hartford,  Conn. 

The  accounting  side  of  the  whole  of  the  undertakings 
examined  was  entrusted  to  myself  and  Mr.  R.  C.  James,  the 
Chief  Accountant  to  the  United  Gas  Improvement  Company,  of 
Philadelphia.  During  the  investigation  all  the  experts  were  in 
constant  communication  and  worked  together,  and  each  under- 
taking was,  as  far  as  possible,  examined  by  them  at  the  same 
time.  This  resulted  in  a  very  minute  comparison  of  American 
and  British  technical  and  accounting  practice.  During  the 
months  we  were  associated  many  friendships  were  formed,  but 


xii  PREFACE 

the  greater  value  of  the  enquiry  to  all  the  experts  engaged  is 
undoubtedly  the  wider  appreciation  of  American  and  British 
practice.  On  the  technical  and  practical  side,  many  hitherto 
diverging  views  were  reconciled  or  reduced  to  an  agreed  mean, 
but  as  regards  the  accounting  and  financial  side  no  such 
difference  was  found.  In  America,  as  in  Great  Britain,  all 
problems  relating  to  large  financial  operations  follow  the  same 
economic  laws,  and  the  only  variations  on  each  side  of  the 
Atlantic  are  due  to  the  actual  conditions  now  existing.  On 
this  side  the  growth  of  large  municipal  undertakings,  concerned 
with  the  provision  of  public  utilities,  has  been  a  gradual  process 
extending  over  many  years  requiring  the  constant  supervision 
of  Parliament.  On  the  other  side  the  municipal  operation  of 
public  utilities  is  still  in  its  infancy,  and  is  therefore  an  urgent 
practical  problem,  as  evidenced  by  the  enquiry,  made  at  very 
considerable  expense,  by  the  National  Civic  Federation  of  New 
York. 

In  connection  with  pure  methods  of  accounting,  it  must  be 
acknowledged  tliat  as  regards  uniformity  the  British  system  is 
not  so  far  advanced  as  the  American.  Standard  methods  have 
there  loujx  been  adopted  for  the  accounts  of  gas  works,  electric 
lighting  undertakings,  and  street  railways,  and  include  not 
only  the  form  of  the  final  Revenue  Account  and  Balance  Sheet, 
but  also  methods  of  analysis  and  book-keeping  in  minute  detail. 
It  is  true  that  in  Great  Britain  standard  forms  of  accounts  for 
gas  works  are  contained  in  the  Gas  Works  Clauses  Act.  The 
Board  of  Trade  have,  under  their  power  in  the  Electric  Lighting 
Acts,  prescribed  standard  forms  of  account  for  electric  lighting 
undertakings.  A  standard  form  of  account  for  tramways  has 
been  drawn  up  by  the  Tramway  Institute  and  the  Institute  of 
Municipal  Treasurers  and  Accountants,  and  as  the  result  of  an 
enquiry  by  a  Government  Departmental  Committee  (report 
dated  1907)  further  amended  standard  forms  have  been 
suggested  for  all  tlie  above-mentioned  utilities.  The  British 
standard  forms,  however,  are  deficient  to  the  extent  that  they 
lack  that  minute  attention  to  detail  which  is  the  characteristic 
of  the  American  systems.  A  standard  form  of  revenue,  or 
profit  and  loss,  account  may  be  very  useful  for  comparing  the 
final  results  of  the  year's  operations,  but  unless  there  is  absolute 
uniformity  of  detail  in  the  items  charged  to  each  head  of 
operating  expense  it  is  impossible  to  make  any  reliable 
comparison  between  the  operating  costs  of  two  or  more  under- 
takings. This  fact  was  very  clearly  shown  in  the  enquiry  by 
the  Commission  in  Great  Britain. 


PREFACE  xiii 

Turning  now  to  the  immediate  subject  of  this  book,  namely, 
the  repayment  of  the  loan  debt  of  local  authorities  and  privately 
owned  undertakings,  the  result  of  the  enquiry  proved  that  as 
regards  the  actual  methods  of  repayment  the  practice  is  the 
same  in  both  countries.  Another  outstanding  feature  of  the 
enquiry  was  the  minute  attention  paid  by  the  legal,  labour  and 
economic  experts  upon  the  Commission  to  the  statutory  and 
other  obligations  imposed  by  Parliament  upon  local  authorities 
in  Great  Britain.  This  was  due  to  the  fact  that  the  municipal 
operation  of  public  utilities  in  this  country  is  a  much  older 
institution  than  in  the  United  States.  The  report  of  the 
Commission  contains  much  valuable  information  in  considerable 
detail  as  to  these  statutory  obligations  and  is  a  useful  resume 
of  a  very  complicated  question,  well  worth  perusal  by  municipal 
experts  in  this  country.  This  book  does  not  in  any  way  attempt 
to  deal  with  these  matters  in  an  exhaustive  manner,  and  they 
are  only  mentioned  in  so  far  as  they  relate  to  the  actual  methods 
of  repayment  of  loan  debt.  In  writing  the  book  I  have  borne  in 
mind  the  results  of  the  enquiry  and  have  so  arranged  it  that  it 
will  apply  to  all  problems  in  whatever  currency.  The  formula 
relating  to  a  geometrical  progression  does  not  recognise  any 
geographical  limits,  consequently  the  methods  based  thereon 
may  be  applied  equally  in  the  United  States,  as  in  Great 
Britain.  Owing  to  the  interest  in  municipal  ownership 
and  operation  in  the  United  States,  the  practical  application  of 
the  statutory  obligations  as  regards  the  repayment  of  loan  debt 
by  British  municipalities,  even  as  briefly  stated,  will,  it  is 
hoped,  be  useful. 

As  far  as  possible  the  terms  used  have  been  chosen  in  order 
to  avoid  any  misunderstanding  on  either  side.  The  word 
"  Corporation  "  in  Great  Britain  denotes  both  a  Local  Authority 
and  a  privately  owned  undertaking.  The  term  "  Local 
Authority "  has  therefore  been  used  throughout,  although  it 
may  not  in  a  few  cases  be  strictly  correct.  It  includes  all 
public  authorities  having  control  of  public  moneys  for  the 
public  good  and  empowered  to  raise  such  moneys  by  way  of  an 
annual  rate  based  upon  an  assessment  of  the  annual  value  of  the 
property.  It  is  not  the  practice  in  Great  Britain  to  levy  a  local 
rate  or  assessment  based  upon  the  capital  value.  All  such  rates 
are  levied  at  so  many  pence  in  the  pound  {£)  sterling  of  annual 
value. 

In  Great  Britain  there  are  several  kinds  of  privately  owned 
companies  or  corporations,  which  may  be  divided  into  two 
groups.     First,  those  in  which  the  liability  of  the  members  is 


xiv  PREFACE 

unlimited,  wliicli,  however,  are  not  numerous.  Secondly,  those 
companies  in  whicli  the  liability  of  the  members  is  limited  to 
the  actual  amount  for  the  time  being  unpaid  upon  the  shares 
held  by  them.  Such  companies  may  be  divided  into  two  classes, 
namely,  those  which  derive  their  powers  of  operation  from,  and 
are  incorporated  by,  special  Act  of  Parliament;  and,  secondly, 
those  incorporated  under  the  Limited  Liability  Acts  (The 
Companies  Acts,  1862 — 13)08).  In  this  book  all  such  companies 
liave  been  included  under  the  generic  term  "  Private  Under- 
takings "  as  distinguished  from  "  Local  Authorities." 

The  capital  of  all  privately  owned  companies  or  undertakings 
in  Great  Britain  is  provided  by  the  members  and  is  raised  in 
various  ways.  In  the  case  of  companies  incorporated  under  the 
Limited  Liability  Acts  the  capital  is  invariably  raised  by  the 
issue  of  a  definite  number  of  shares  of  a  uniform  nominal  value, 
now  as  a  general  rule  of  £1  each.  In  the  case  of  companies 
incorporated  by  special  Act  of  Parliament  the  capital  is  some- 
times raised  by  the  issue  of  shares,  similar  to  companies  incor- 
porated under  the  Limited  Liability  Acts,  but  often  by  the 
issue  of  stock.  All  capital  raised  by  all  privately  owned 
undertakings,  whether  by  shares  or  stock,  may  be  issued  with 
certain  defined  priorities  or  preferences  both  as  to  dividend  and 
also  as  to  repayment  upon  the  final  winding  up  of  the  company. 
In  all  cases  considered,  the  capital  provided  by  the  members 
of  a  privately  owned  undertaking  has  been  included  under  the 
generic  term  of  "  Share  Capital  or  Stock." 

The  term  "  Loan  Debt  "  has  been  used  to  denote  all  moneys 
borrowed  by  a  public  authority  or  private  undertaking  and 
secured  by  way  of  mortgage  upon  the  assets,  including  in  the 
term  assets  the  power  of  a  local  authority  to  levy  a  rate  or 
assessment.  In  the  case  of  a  private  undertaking  such  loan 
debt  is  always  repayable,  on  a  winding  up,  in  priority  to  the 
share  capital  or  stock,  and  may  or  may  not  be  repayable,  during 
the  life  of  the  undertaking,  by  means  of  a  sinking  fund  to  be 
built  up  out  of  the  annual  profits.  In  the  case  of  all  public 
authorities  in  Great  Britain,  Parliament  now  invariably  imposes 
the  obligation  to  repay  such  loan  debt  by  means  of  a  sinking 
fund,  or  other  alternative  method,  Avithin  a  period  having  a 
more  or  less  definite  relation  to  the  life  of  the  asset  created  out 
of  tlie  loan,  and  such  annual  provision  for  repayment  almost 
witliout  (xccption  operates  immediately  upon  the  borrowing  of 
the  money,  and  is  charged  against  the  annual  rate  or  assessment 
levied  by  the  h)cal  authority.  In  the  case  of  a  loan  raised  to 
ptovido  works  of  a  revenue-earning  character,  such  as  gas  works, 


PREFACE 


XV 


etc.,  the  annual  redemption  cliarges,  as  well  as  tlie  interest  upon 
tlie  loan,  are  charged  against  the  profits  of  the  undertaking, 
and  any  deficiency  is  made  good  out  of  the  annual  rate  or 
assessment  levied  upon  the  whole  of  the  community. 

The  foregoing  remarks  will  sufficiently  explain  the  terms 
used  in  the  book,  but  if  fuller  details  are  required  the  reader  is 
referred  to  the  Financial  Appendix  to  the  Report  of  the  Com- 
mission (Vol.  II,  Part  II,  p.  628),  prepared  by  Mr.  James  and 
myself. 

E.H.T. 
Manchester  (Enr/.), 

November,  1912. 


SUMMARY    OF    SECTIONS. 


PAGE 

I.   Mathematical  Principles. 

Chapters  II  to  X       19 

II.  The  Methods  of  Repayment  of  Loan  Debt. 

Chapters  XI  to  XIII      107 

III.  The  Annual  Instalment. 

Chapters  XIV  to  XXI      143 

IV.  The  Annual  Increment, 

Chapters  XXII  to  XXVII     257 

V.  The  Date  of  Borrowing. 

Chapters  XXVIII  to  XXX     343 

VI.   The  Life  of  the  Asset  and  its  Relation  to 

the  Redemption  Period. 

Chapter  XXXI     375 

VII.   The  Equation  of  the  Period  of  Repayment. 

Chapter  XXXII     387 

VIII.  The  Equation  of  the  Incidence  of  Taxation. 

Chapters  XXXIII  to  XXXV     407 


CONTENTS. 


CHAP.  P'^*^'^ 

I.    Introduction "      i 


SECTION    I. 
Mathematical  Principles. 


CHAP. 

II.    Logarithms 


III.  Simple   and    Compound   Interest 28 

Compound  interest  a  geometrical  progression. 
Derivation  of  the  formula, 

A  =  PR^ 

relating  to  compound  interest,  from  the  algebraical  formula, 

I  —  ar""^ 
relating    to    a    geometrical    progression.      Explanation    of 
terms.     Difference  between  the  amount  of  /,i,  and  of  £1  per 
annum,  at  the  end  of  one  year.     "  Present  Value  "  com- 
pared with  "  Practical  Discount." 

IV.  Compound  Interest  as  Applied  to  a  Sum  of  Money  -        -    36 

Table  I.  The  amount  of  £1  in  any  number  of  years. 
The  formula,  A  =  P  RN  and  rules  deduced  therefrom. 
Calculation  by  the  arithmetical  method.  Compilation  of 
tables.  Thoman's  method  and  formula.  Author's  Standard 
Calculation  form.  No.  i. 

V.  Compound  Interest  as  Applied  to  a  Sum  of  Money  {contd.)  -    42 

Table  II.  The  present  value  of  £1  due  at  the  end  of  any 
number  of  years. 

^  ~   R^ 

Derivation  of  the  formula  and  rules  deduced  therefrom. 
Tables  of  ratios  and  logs,  of  R,  and  r;  calculations  for 
periods  other  than  years.  Thoman's  method  and  fornnila. 
Author's  Standard  Calculation  form,  No.  2. 


XX  CONTENTS 

chap,  page 

\1.    Compound  Interest  as  Applied  to  an  Annual  or  other 

Periodic  Payment '5° 

Table  III.     The  amount  of  £i  per  annum  in  any  number  of 
years. 


'C^) 


M  =  A2/( 

Derivation  of  the  formula  and  rules  deduced  therefrom. 
Thoman's  method  and  formula.  Author's  vStandard  Calcu- 
lation form,  No.  3. 


VII.    Compound  Interest  as  Applied  to  an  Annual  or  other 

Periodic  Payment  {coutd.) 62 

Table   IV.     The   present   value  of  £1   per  annum   for  any 
number  of  years. 


P  =  Ay 


Derivation  of  the  formula  and  rules  deduced  therefrom. 
Thoman's  method  and  formula.  Author's  Standard  Calcu- 
lation form,  No.  4. 


VIII.    Compound  Interest  as  Applied  to  an  Annual  or  other 

Periodic  Payment  (contd.)        -        -        -        -        -        ■    67 

Table    V.     The    anuuit}-    which   £1    will    purchase    for    an}- 
number  of  3'ears,  or  of  which  £1  is  the  present  value. 


A'/-(|^) 


Derivation  of  the  formula  and  rules  deduced  therefrom. 
Thoman's  method  and  formula.  Author's  Standard  Calcu- 
lation form.  No.  5. 


IX.  Tii()^l\n's  LoGARniiMic  Tables  of  Compound  Interest  and 

Annuities    -        -        -        -        -        -        -        -        -        -73 

Explanation  of  the  symbols  used   in-   Thoman,   and   their 
relation  to  the  abo\e  tallies  and  formula;. 

X.  Standard  Calculation   Forms,   prepared   by  the   Author, 

relating  To  TIIK  ABOVE  TABLES,  WITH  E-XPLANATIONS  AND 
INSTRUCTIONS  AS  TO   MAKING  THE  CALCULATIONS  -  -     8o 


CONTENTS 


SECTION    II. 
The  Methods  of  Repayment  of  Loan  Debt. 

CHAP.  PAGE 

XI.  The  Repayment  of  the  Loan  Debt  of  Local  Authorities 

AND   commercial  AND   FINANCIAL   UNDERTAKINGS  -  -  109 

Alternative  methods  allowed  by  the  Public  Health  Act,  1S75, 
and  other  Acts.  Comparison  of  methods  as  regards  the 
actual  repayment  to  the  lender  and  the  annual  charge 
against  revenue  or  rate. 

The  Instalment  Method,  by  equal  annual  instalments  of 
principal  repaid  to  the  lender.  vStatement  showing  the  final 
repayment  of  the  loan. 

XII.  The  Repayment  of  the  Loan  Debt  of  Local  Authorities, 

ETC.   (contd.) -        -        -114 

The  Annuity  Method,  by  an  equal  annual  instalment  of 
principal  and  interest  combined,  repayable  to  the  lender. 
Methods  of  calculation.  FormuhTC  and  rules  deduced  there- 
from. vStatement  showing  the  final  repayment  of  the  loan. 
Comparison  with  the  sinking  fund  iiistal;nent,  and  the 
equal  annual  instalment  of  principal  only. 

XIII.  The  Repayment  of  the  Loan  Debt  of  Local  Authorities, 

ETC.    (contd.)         -        -        - 126 

The  Sinking  Fund  Method,  by  setting  aside  and  accumu- 
lating an  equal  annual  instalment  to  provide  the  principal 
only  at  the  end  of  the  redemption  period. 

1.  The  Accumulating  Sinking  Fund.  Methods  of  calcula- 
tion. Formula  and  rules  deduced  therefrom.  Statement 
showing  the  final  repayment  of  the  loan. 

Author's  Standard  Calculation  form.  No.  3X. 

2.  The  Non-accumulating  vSinking  Fund,  compared  with 
the  instalment  and  accumulating  sinking  fund  methods. 
Statement  showing  the  final  repayment  of  the  loan.  State- 
ment showing  the  comparison  between  the  three  methods 
already  described. 


CONTENTS 


SECTION   III. 
The  Annual  Instalment- 

CHAP.  PAGE 

XI\'.     Sinking  Fund  Problems  in  general,  relating  to  :      -        '  145 

1.  The  amount  in  tlie  fund. 

2.  The  rate  per  cent, 

(a)  of  income  to  be  received  upon  the  present  investments 

representing  the   fund. 

(b)  the  future  rate  of  accumulation. 

3.  The  redemption  period. 

4.  The    rate    per    cent,     and    the    redemption    period    in 

combination. 
Definition  and  explanation  of  the  terms  "  present  invest- 
ments "  and  "  annual  increment." 

XV.  Sinking  Fund  Problems  relating  to  the  amount  in  the 

Fund.    A  deficiency  in  the  fund   -        -        -        ■        -  154 
Calculation  of  a  t3^pical  fund  to  be  used  as  an  example. 
How  a  deficiency  ma}'  arise,  and  how  it  may  be  ascertained. 
Summary  of  the  methods  of  making  the  adjustment  in  the 
annual  instalment. 

XVI.  Sinking  Fund  Problems  relating  to  the  amount  in  the 

Fund.     A  deficiency  in  the  fund  {contd.)    -        -        -  171 

Variation  i.  To  be  corrected  b}-  an  additional  annual 
instalment  to  be  set  aside  during  the  whole  of  the  unexpired 
portion  of  the  repayment  period. 

Variation  2.  To  be  corrected  by  an  additional  annual 
instalment  to  be  set  aside  during  the  earlier  part  only  of 
the  unexpired  portion  of  the  repayment  period. 

XVII.  vSiNKiNG  Fund  Problems  relating  to  the  amount  in  the 

Fund  (contd.).     A  surplus  in  the  fund  -        -        -        -  186 

Variation  i.  Arising  in  consequence  of  an  excessive  past 
accumulation  of  tlie  fund. 

\'ariation  2.  Arising  in  consequence  of  the  payment  into 
the  fund  of  the  proceeds  of  sale  of  part  of  the  assets 
representing  the  securit}^  for  the  loan,  or  a  realised  jn'ofit 
upon  the  sale  of  an  investment  representing  the  fund. 


CONTENTS  3^^"^ 

PAGE 
^YVIIl.    Sinking  Fund  Problems  relating  to  the  amount  in  the 

Fund  (contd.)      --'''''  ^99 

A  surplus  in  the  fund  of  a  coniniercial  or  financial  under- 
taking arising  on  the  withdrawal  of  part  of  the  loan  from 
the  operation  of  the  fund,  owing  to  the  conversion  of  such 
part  of  the  loan  into  ordinary  share  capital  or  stock  of  the 
undertaking. 

Variation  3.  In  which  the  original  annual  instalment  was 
found  by  calculation  based  upon  a  specified  period  of 
repayment  and  rate  of  accumulation. 

Variation  4.  In  which  the  original  annual  instalment  is 
a  stated  sum,  and  is  not  based,  except  in  a  general  way, 
upon  any  period  of  repayment  or  rate  of  accumulation. 

XIX     Sinking  Fund  Problems  relating  to  the  rate  per  cent. 
OF  income  upon  the  present  investments  representing 

THE  amount  in  THE  FUND  ;  AND  .^LSO  THE  FUTURE  RATE  OF 
ACCUMULATION    OF    THE    FUND.        -  "  "  "  "  '^^S 

Variation  A.  In  ^vhich  there  is  a  variation  in  the  rate  of 
accumulation  without  any  variation  in  the  rate  of  income 
upon  the  present  investments,  or  in  the  period  of  repayment. 

XX.    Sinking  Fund  Problems  relating  to  the  rates  per  cent. 

OF   INCOME  AND   ACCUMULATION    (COUtd.)         "  '  "  "236 

Variation  B.     In  which  there  is  a  variation  in  the  rate  of 
income  upon  the  present  investments  without  any  variation 
iu  the  rate  of  accumulation,  or  in  the  period  of  repayment. 
The  subject  is  further  considered  in  Chapter  XXVII. 

XXI.    Sinking  Fund  Problems  relating  to  the  rates  per  cent. 

OF   INCOME  AND   ACCUMULATION    [COntd.)         -  "  '  "247 

Variation  C.  In  which  there  is  a  variation  in  the  rate  of 
accumulation,  and  also  in  the  rate  of  income  upon  the 
present  investments,  but  without  any  variation  m  the 
period  of  repayment. 


xxiv  CONTENTS 

Section   IV. 
The  Annual  Increment. 

CHAP.  PAGE 

XXII.  Sinking  Fund  Problems  relating  to  the  rate  per  cent. 

of  ACCUMULATION 259 

The  above  variations  further  considered,  namel}-,  a  variation 
in  the  rate  of  accumulation  with  or  without  a  variation  in 
the  rate  of  income  upon  the  present  investments,  but 
without  any  variation  in  the  period  of  repayment. 
Comparison  of  the  results  already  obtained  in  terms  of  the 
annual  instalment,  with  those  obtained  by  means  of  the 
annual  increment,  and  the  varying  rates  of  accumulation. 

XXIII.  Sinking  Fund  Problems  relating  to  the  rate  per  cent. 

OF  accumulation  (contd.)  -        - 277 

Derivation  of  a  rule  and  formula  relating  to  a  variation  in 
the  rate  per  cent,  of  accumulation  based  upon  the  foregoing 
results,  by  the  annual  increment   (ratio)   method. 

XXIV.  Sinking  Fund  Problems  relating  to  the  redemption 

PERIOD  -----------  283 

The  deductive  method  of  finding  the  amended  annual 
instalment  due  to  a  variation  in  the  period  onlj-.  The 
annual  increment  (ratio)  method. 

XXV.  Sinking  Fund   Problems   rel.\ting  to  the  redemption 

PERIOD   (contd.)    -        -        - 295 

Derivation  of  a  rule  and  formula  relating  to  a  variation  in 
the  period  of  repayment  based  upon  the  foregoing  results, 
by  the  annual  increment   (ratio)   method. 

XXVI.  Sinking  Fund  Problems  rel.ating  to  the  rate  per  cent. 

OK    accu.mulation    and    the    redemption    period    in 
combination         -.- 200 

The  deductive  method  of  ascertaining  the  amended  annual 
instalment  due  to  a  variation  in  both  the  above  factors  in 
combination. 

Derivation  of  a  rule  and  formula  relating  to  a  dual  variation 
of  this   nature   based  upon   the   foregoing   results,   bj^   the 
annual  increment  (ratio)  method. 


CONTENTS  XXV 

CHAP.  PAGE 

XXVII.    Sinking   Fund   Problems   relating   to  the   rate   per 

CENT.   OF  income  UPON  THE  PRESENT  INVESTMENTS  REPRE- 
SENTING   THE   FUND        -------  -322 

(In  continuation  of  Chapter  XX)  but  in  which  the  rate  of 
income  yielded  by  such  investments  is  not  uniform  during 
the  whole  of  the  unexpired  portion  of  the  repayment  period. 

1.  In  which  the  future  variation  in  the  rate  of  income  is 
known,  and  is  definite  both  as  to  time  and  amount. 

2.  In  which  the  future  variation  in  the  rate  of  income  is 
anticipated,  but  is  uncertain  both  as  to  time  and  amount. 


SECTION  V. 

The  date  of  Borrowing,  and  its  relation  to  the  Redemption 
Period. 

WITHOUT      ANY      COMPLICATION      AS      REGARDS      THE      LIFE      OR 

DURATION  OF  CONTINUING  UTILITY   OF   THE   ASSET  CREATED   OUT 

OF  THE  LOAN. 

XXVIII.  Loan  borrowed  over  several  years  in  one  sum  in 

EACH   year,    each    YEAR'S    BORROWINGS    BEING    REPAYABLE 

IN  A  PRESCRIBED  PERIOD  FROM  THE  D.\TES  OF  BORROWING  -  345 

1.  By  means  of  one  sinking  fund  only. 

2.  By  separate  sinking  funds  for  each  year's  borrowings. 

XXIX.  Loan  borrowed  over  several  years  in  one  sum  in  each 

YEAR,  REPAYABLE  IN  ONE  SUM  ON  A  CERTAIN  SPECIFIED  DATE  354 

1.  Where  the  date  of  repayment  is  known  at  the  time  the 
money  is  borrowed. 

2.  Where  the  date  of  repayment  is  fixed  after  the  sinking 
fund  has  been  in  operation  for  a  number  of  years,  and  an 
adjustment  of  the  fund  is  required. 

XXX.  Loan   borrowed   in  one   or  more  years,   in   varying 

AMOUNTS  and  AT  VARIOUS  DATES  IN  EACH  YEAR,  AND  IT  IS 
REQUIRED  THAT  THE  REVENUE  OR  RATE  ACCOUNT  OF  EACH 
YEAR  SHALL  BE  CHARGED  WITH  A  PROPORTIONATE  PART  OF 
THE   ANNUAL   SINKING   FUND    INSTALMENT      -  -  "  -  365 


CONTENTS 


SECTION   VI. 

CHAP.  PAGE 

XXXI.    The  Life  of  the  Asset,   and  its  relation  to  the 

Redemption  Period    -        -  -        -        -        -  377 


SECTION    VII. 
XXXII.         The  Equation  of  the  Period  of  Repayment 

OF    LOANS     REPAYABLE     AT     VARIOUS     DATES,     WHICH     ARE 
REQUIRED  TO  BE  REDEEMED  ON  ONE  UNIFORM  DATE  -    389 

1.  Where  the  loans  are  authorised  in  respect  of  outlays  of 
varying  character,  each  having  a  different  life  or  period  of 
continuing  utility  and  consequent  repa3nnent. 

2.  Where  the  necessity  to  find  the  equated  period  of  repay- 
ment arises  on  the  consolidation  of  existing  loans. 

The  arithmetical  method  of  finding  the  equated  period 
known  as  the  "  equation  of  payments."  The  true  or 
mathematical  method.  The  error  in  the  generally  adopted 
arithmetical  method. 


CONTENTS 


SECTION   VIII. 
The  Equation  of  the  Incidence  of  Taxation. 

CHAP.  PAGE 

XXXIII.  Comparison  of  the  total  annual  loan  charges  to 

REVENUE  OR  RATE  BEFORE,  AND  AFTER,  THE  EQUATION  OF 
THE  PERIOD  OF  REPAYMENT,  SHOWING  THE  UNEQUAL 
INCIDENCE  OF  TAXATION,  IF  THE  ANNUAL  INSTALMENT,  AND 
INTEREST  UPON  THE  TOTAL  LOAN,  BE  SPREAD  EQUALLY  OVER 
THE  EQUATED   PERIOD -  -  -  ^09 

XXXIV.  The  Annual  Instalment 418 

The  various  methods  of  adjusting  the  annual  charges  to 
revenue  or  rate  during  the  equated  period   in  proportion 

to  the  life  or  duration  of  continuing  utility  of  the  asset 
created  out  of  the  loan,  viz.  :  — 

By  charging  the  revenue  or  rate  account  of  each  year  of  the 
equated  period  with  the  annual  instalment  chargeable 
against  each  year  before  equation,  and  in  addition  thereto 
a  supplementary  annual  instalment  :  — 

(a)  to  be  spread  equally  over  the  equated  period ;  or 

(b)  to  be  proportionate  year  by  year  to  the  annual  instal- 
ments before  equation. 


XXXV.    Interest  upon  the  Loan  ------       -436 

The  method  of  adjusting  the  annual  interest  charges  to 
revenue  or  rate  during  the  equated  period  in  proportion  to 
the  life  or  duration  of  continuing  utility  of  the  asset  created 
out  of  the  loan  :  — 

By  charging  the  revenue  or  rate  account  of  each  year  of  the 
equated  period  with  the  annual  amount  of  interest  payable 
before  equation,  and  in  addition  thereto  a  supplementary 
annual  amount  proportionate  year  by  year  to  the  annual 
interest  charges  before  equation.  A  general  Summary  of 
the  results  obtained  in  Chapters  XXXIII,  XXXIV,  and 
XXXV,  illustrated  by  diagrams. 


Introduction 


CHAPTER  I. 

INTRODUCTION. 

This  book  is  the  outcome  of  many  years'  professional  work 
ill  connection  with  the  accounts  of  municipal  corporations,  other 
local  authorities,  and  privately  owned  commercial  and  financial 
undertakings.  It  deals  only  with  the  loan  debt  of  such  public 
authorities  and  private  undertakings,  and  includes,  in  addition 
to  the  actual  borrowing  and  repayment  of  the  loan,  the 
numerous  problems  which  arise  in  connection  with  the  Sinking 
Fund,  relating  to  the  amount  in  the  fund,  the  rate  of  accumula- 
tion, and  the  period  of  repayment.  The  concluding  sections 
contain  chapters  upon,  (1)  the  relation  between  the  life  of  the 
asset  and  the  period  of  repayment;  (2)  the  methods  of  finding 
the  equated  period  of  repayment ;  and  (3)  the  equation  of  the 
incidence  of  taxation  after  the  equation  of  the  period,  both  as 
regards  the  annual  instalment  and  interest  upon  the  loan. 
In  the  last  three  chapters  this  difficult  subject  is  treated  in  an 
exhaustive  manner. 

Subject  Matter.  The  book  does  not  pretend,  in  any  way, 
to  be  a  treatise  upon  the  law  relating  to  the  loan  debt  of  local 
authorities,  or  to  give  full  particulars  of  the  various  statutory 
obligations,  as  regards  repayment,  imposed  by  Parliament ;  nor 
does  it  include  a  full  statement  of  the  general  practice  of 
Parliament  and  the  government  departments  having  control  of 
such  matters.  All  such  statutory  obligations  are  of  a  very 
variable  nature  and  are  contained  in  many  general  and  special 
Acts  of  Parliament  and  provisional  orders  of  the  Local  Govern- 
ment Board.  The  general  practice  cannot  be  said  to  be  based 
upon  any  well-defined  principles,  but  it  should  be  stated  that 
the  Local  Government  Board  endeavours,  as  far  as  it  is 
empowered,  to  impose  a  uniform  system,  especially  as  to  the 
periods  to  be  allowed  for  the  repayment  of  loans  raised  for 
public  works  having  longer  or  shorter  periods  of  duration  or 
continuing  utility. 

Although  the  actual  practice  varies  considerably  in  detail, 
the  methods  to  be  adopted  in  the  solution  of  the  various 
problems  all  follow  certain  well-defined  mathematical  rules; 
consequently  the  primary  object  is  to  demonstrate  briefly  the 
mathematical  principles  involved  and  afterwards  to  apply  such 
c 


2  REPAYMENT    OF    LOCAL   AND    OTHER   LOANS 

principles  to  a  number  of  typical  examples.  As  between  the 
mathematical  and  practical  sides  of  the  subject,  preference  has 
been,  and  must  necessarily  be,  given  to  the  mathematical 
portion,  because  this  is  definite,  and  may  be  exactly  stated. 
The  practical  variations  from  the  ideal  mathematical  conditions 
are  so  numerous  that  only  typical  examples  have  been  con- 
sidered, although  every  effort  has  been  made  to  include  all  the 
principal  problems  which  are  likely  to  arise. 

Mathematical  Principles.  Since  all  problems,  relating  to 
the  future  redemption  or  repayment  of  a  present  loan,  to  be 
spread  over  a  period  of  years,  involve  questions  of  compound 
interest,  it  is  first  necessary  to  investigate  the  mathematical 
principles  governing  the  annual  or  other  periodic  accumiilation 
of  a  present  sum  of  money,  and  also  of  a  sum  of  money  payable 
or  receivable  at  the  end  of  each  of  a  number  of  equal  and 
definitely  recurring  periods  of  time.  All  such  problems  follow- 
the  algebraical  rule  relating  to  a  geometrical  progression,  as 
distinguished  from  the  rule  relating  to  an  arithmetical  progres- 
sion which  concerns  only  simple  interest.  The  primary  object 
therefore  is  to  convert  the  simple  algebraical  formula  relating 
to  a  geometrical  progression  into  a  formula  which  may  be 
applied  to  the  subject  matter  of  this  book,  namely,  compound 
interest.  Simple  interest  is  purely  a  matter  of  arithmetical 
calculation  and  does  not  arise  in  any  way  in  the  problems  to 
be  discussed.  On  the  other  hand,  compound  interest  involves 
a  mathematical  method  of  calculation  and  affects  all  problems 
which  will  be  hereafter  considered. 

The  algebraical  formula  relating  to  any  geometrical  pro- 
gression, as  regards  the  last  and  first  terms  in  any  series  of 
numbers  is :  — 

and  this  formula  may  be  converted  into  a  standard  formula 
relating  to  the  accumulation  of  a  sum  of  money  now  in  hand, 
namely  :  — 

A=PEN 

as  described  in  Chapter  III.  The  derivation  of  the  formula 
relating  to  the  accumulation  of  an  annuity  or  other  periodic 
payment,  namely  :  — 

M  =  A,/(?f^) 

is  described   in  Chapter  YI. 


INTRODUCTION  3 

FoRMUL.E  AND  SYMBOLS.  The  svmbols  adopted  by  the  author 
differ  somewhat  from  those  given  in  the  books  on  algebra,  and 
in  other  mathematical  works.  They  have,  however,  been 
chosen  after  much  consideration  in  order  to  afford  some 
indication  of  the  factors  they  represent.  The  very  full  treat- 
ment which  is  given  to  the  various  formulae  is  due  to  the  fact 
that  they  are  indispensable  if  a  calculation  has  to  be  made  at 
any  rate  per  cent,  not  included  in  the  published  tables  of 
compound  interest.  A  detailed  explanation  of  the  symbols  and 
formulae  is  contained  in  Chapter  X,  dealing  with  the  standard 
calculation  forms  prepared  by  the  author. 

Logarithms.  Throughout  the  book,  the  method  of  calcula- 
tion is  entirely  by  logarithms,  since  any  attempt  to  arrive  at 
the  results  by  arithmetical  methods  would  involve  a  serious 
waste  of  time,  and  a  greater  liability  to  errors  in  computation. 
The  use  of  logarithms  is  fully  explained  in  the  usual 
arithmetical  works,  and  also  in  the  introduction  to  most  of  the 
published  tables  of  logarithms,  but  a  short  chapter  (No.  II)  has 
been  included  in  order  to  make  the  book  self-contained.  There 
is  not  anything  at  all  difficult  in  the  use  of  a  table  of  logs. : 
which  is  merely  a  very  much  neglected  "  ready-reckoner." 
There  are  several  good  tables  giving  seven-figure  logarithms 
of  the  numbers  from  1  to  108000. 

Mathematical  Tables.  There  are  many  published  tables 
of  compound  interest,  which  may  be  used  to  facilitate  the 
various  calculations,  and  which  may  be  divided  into  three 
groups,  namely  :  — 

(1)  Tables  giving  the  actual  values  of  £1,  and  of  £1  per 
annum,  for  various  periods  at  stated  rates  per  cent,  per  annum. 
These  tables  are  valuable  in  proportion  to  the  number  of  rates 
per  cent,  for  which  the  actual  values  are  given.  In  using  all 
such  tables,  a  table  of  logs,  is  also  required.  In  England,  the 
one  most   generally  used   is   known   as   Inwood's   Tables    (21st 

'  edition,  1880),  and  the  new  edition  by  Schooling  (1899).* 

(2)  Tables  in  tvhich  the  actual  values  are  not  given,  but 
ivhich  contain  their  logarithndc  equivalents.  The  tables  of 
M.  Fedor  Thoman  are  of  this  type,  and  they  are  especially 
valuable,  because  they  are  worked  out  for  many  intermediate 
rates  per  cent,  not  given  in  Inwood's  and  other  similar  tables 
of   compound    interest,   and    also   because   they  enable    one   to 

*  In   America,    Tables  of   Compound    Interest,    Discount,   Sinking    Funds, 
Annuities,  etc.,  by  Charles  E.  Sprague. 


4  REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

dispense  with  a  table  of  logs.,  except  as  regards  the  actual  sums 
of  money  involved  in  the  calculation.  They  are  particularly 
useful  because  all  values  are  reduced  to  two  factors  only, 
namely,  E^,  and  a",  by  various  combinations  of  which  all  the 
calculations  may  be  made.  The  derivation  and  use  of  these 
tables  are  fully  explained  in  Chapter  IX. 

(3)  Tables  icorked  out  on  the  "ready-reckoner"  principle, 
giving,  for  example,  the  sinking  fund  instalments,  or  the  equal 
annual  instalments  of  principal  and  interest  combined,  for  £1, 
and  multiples  of  £1,  for  various  periods  of  years,  at  various 
rates  per  cent.  Such  tables  may  be  very  useful  to  some,  but 
they  have  not  any  educational  value  whatever,  and  it  is 
doubtf  vil  if  they  effect  any  actual  saving  of  time  when  compared 
with  the  other  types  of  tables,  especially  Thoman's,  which  are, 
when  possible,  always  used  by  the  author.  The  practical  value 
of  tables  of  this  kind  is  limited  by  the  number  of  rates  per  cent, 
actually  worked  out  in  detail,  and  the  same  applies  to  Inwood's 
and  Thoman's  tables. 

If  a  problem  be  required  to  be  worked  out  at  any  rate  per 
cent,  not  given  in  any  published  table,  such  problem  is 
impossible  of  solution  by  anyone  not  acquainted  with  the 
mathematical  principles  wpon  which  all  such  tables  are  based. 
The  object  throughout  has  been  to  reduce  these  mathematical 
principles  to  the  very  simplest  form,  and  to  give  such  minute 
instructions,  and  to  provide  such  standard  forms  of  calculation, 
that  anyone  acquainted  with  the  ordinary  rules  of  arithmetic, 
and  the  use  of  a  table  of  logs.,  may  obtain  the  result  required. 

Standard  Calculation  Foems.  A  special  feature  of  the 
book  is  the  series  of  standard  forms,  which  have  been  specially 
prepared  by  the  author,  and  by  means  of  which  all  the  calcula- 
tions in  the  book  have  been  made.  They  are  fully  described  in 
Chapter  X.  The  advantage  of  using  these  forms  is  that  one 
or  all  of  the  three  methods  given  on  each  form  may  be 
adopted ;  and  it  is  generally  advisable  to  make  the  calculation 
in  two  ways  in  order  to  prove  the  accuracy  of  the  result  and 
also  to  avoid  any  possible  error  due  to  a  misprint  in  the  mathe- 
matical table  used.  The  three  methods  shown  in  each  form 
are  :  — 

A.  by  the  mathematical  formula. 

B.  by  tlie  published  tables  of  com])Ound  interest. 

C.  l)y  Thoman's  Logarithmic  Tables. 

and  ill  ;il1  cases  tlio  calculations  are  made  by  logarithms.     The 
arithmetical  method,  based  iipon  the  published  tables,  is  subject 


INTRODUCTION  5 

to  error,  and  is  therefore  unreliable.  A  supply  of  these  forms 
is  invaluable  to  anyone  requiring  to  make  many  calculations 
of  this  nature,  owing  to  their  uniformity  and  also  because  they 
avoid  any  reference  as  to  the  particular  method  to  be  adopted. 
The  formula,  after  a  time,  suggest  the  method.  As  a  general 
rule  the  author  uses,  in  the  first  instance,  method  C,  by 
Thoman's  Tables,  as  being  the  shorter,  and  also  because  these 
tables  include  a  greater  number  of  fractional  rates  per  cent., 
than  the  ordinary  published  tables  of  compound  interest.  The 
factors  being  expressed  in  their  log.  values,  a  reference  to  the 
log.  table  is  saved.  The  result  is  generally  proved  by  logs,  by 
method  B,  using  the  ordinary  published  tables  of  compound 
interest.  In  very  few  cases  is  it  necessary  to  use  method  A,  by 
formula,  when  the  rate  per  cent,  is  worked  out  in  Thoman's  or 
other  tables,  but  where  the  calculation  is  required  at  a  rate  per 
cent,  not  given  in  either  table,  method  A,  by  formula,  is  the 
only  one  available.  It  is  therefore  necessary  to  become  fully 
acquainted  with  the  method  by  formula,  and  to  use  it  to  prove 
the  result  obtained  by  Thoman's  method  C,  where  it  cannot  be 
proved  by  method  B,  owing  to  the  fact  that  the  particular  rate 
per  cent,  is  not  included  in  the  table  available.  When  it  is 
required  to  ascertain  the  number  of  years  with  accuracy,  the 
use  of  the  formula  is  imperative,  and  the  same  applies  to 
problems  in  which  the  rate  per  cent,  is  required.  Yery 
minute  instructions  as  to  the  use  of  the  forms  are  given  in 
Chapter  X,  which  contains  also  ten  standard  forms  by  which  to 
ascertain  the  rate  per  cent,  or  the  number  of  years. 

Pro  forma  Accounts  of  Sinking  Funds.  Throughout  the 
book  the  author  has  repeatedly  laid  great  stress  upon  the 
supreme  importance  of  following  up  the  original  calculation  of 
the  annual  instalment  by  at  once  preparing  a  pro  forma  account 
showing  year  by  year,  how  the  fund  should  accumulate  until 
maturity.  To  make  these  accounts  fully  answer  their  object 
they  should  be  copied  into  a  book  kept  solely  for  the  purpose 
of  preserving  a  permanent  record  of  all  such  accounts,  and  not 
in  the  current  ledger.  This  course  will  save  endless  trouble  in 
future  years.  If  any  adjustment  be  made  in  the  fund  at  any 
future  time  an  amended  pro  forma  account  should  be  prepared 
and  a  reference  made  to  the  original  account.  If  a  copy  of  each 
calculation  be  forwarded  to  the  Local  Government  Board  it 
will  materially  assist  the  officials  and  simplify,  if  it  does  not 
entirely  avoid,  much  subsequent  correspondence  between  the 
Board   and  the  local   authority.     Several  pro   forma   accounts 


6  REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

have  been  prepared  relating  to  examples  given  in  the  book,  not 
only  with  regard  to  a  normal  sinking  fund,  but  also  as  to  an 
adjustment  of  the  fund  due  to  a  variation  in  the  period  of 
repayment,  the  rate  of  accumulation  and  the  income  from 
investments.  It  is  in  such  cases  of  adjustment  that  the  provision 
of  an  account  of  this  nature,  showing  the  effect  of  the  various 
changes  until  maturity,  becomes  particularly  valuable. 

The  Repaymext  of  Loan  Debt.  Having  in  the  earlier 
chapters  described  the  methods  of  ascertaining  the  working 
formulse  and  rules  relating  to  the  various  classes  of  calculations, 
actual  problems  are  next  considered,  beginning  with  the  re- 
payment of  the  loan  debt  of  local  authorities,  taking  as  a  basis 
the  three  alternative  methods  laid  down  in  Sec.  234  of  the 
Public  Health  Act,  1875.  This  section  is  a  very  concise 
statement  of  such  methods  of  repayment,  especially  when 
supplemented  by  the  non-accumulating  sinking  fund  first 
mentioned  in  the  model  clauses  inserted  by  the  Local  Govern- 
ment Board  in  provisional  orders  about  the  year  1893  and 
which  have  since  been  applied  to  many  special  Acts.  The 
effects  of  the  methods  above  mentioned  are  then  fully  discussed 
both  as  regards  the  lender  and  the  rate  or  revenue  account  of 
the  undertaking,  illustrated  by  examples  worked  out  in  detail. 

These  three  alternative  methods  apply  equally  to  the  re- 
payment of  the  debt  of  privately  owned  commercial  and 
financial  undertakings,  although  the  conditions  in  such  cases 
are  much  more  elastic  and  variable  than  is  the  case  with  local 
authorities.     This  is  fully  discussed  in  Chapter  XIII. 

Problems  eelatixg  to  Sixkixg  FrxDS.  The  remainder  of 
the  book  is  occupied  by  the  discussion  of  actual  problems  relating 
to  sinking  funds  proper,  since  the  instalment  and  annuity 
methods  do  not  involve  the  accumulation  of  any  such  fund  but 
provide  for  the  actual  periodical  repayment  to  the  lender.  Very 
few  complications  are  likely  to  arise  in  the  case  of  the  instalment 
method,  and  any  variations  in  the  annuity  method  will  follow 
the  general  rules  as  to  a  simple  annuity.  Such  problems 
concern  the  amount  in  the  fund  at  any  time,  the  rate  of 
accumulation  of  the  fund,  the  rate  of  income  to  be  received 
upon  the  present  investments  representing  the  fund,  the  period 
of  repayment,  or  a  combination  of  any  or  all  of  these  factors. 

The  amount  in  the  fund  at  any  time  may  be  the  correct 
calculated  amount  which  should  stand  to  the  credit  of  the  fund, 
or  may  vary  therefrom,  resulting  in  a  deficiency  or  a  surplus. 


INTRODUCTION^  7 

A  deficiency  in  the  fund  may  be  due  to  a  fall  in  value,  or  a  loss 
upon  the  realisation,  of  an  investment  representing  the  fund, 
but  may  also  be  caused  by  the  accumulation  of  many  minor  past 
deficiencies  in  the  annual  income  received  from  the  investments ; 
and,  although  it  does  not  now  often  occur,  may  be  due  to  a 
deficiency  in  the  annual  instalments  set  aside  in  past  years. 
Cases  have  occurred,  within  the  knowledge  of  the  author,  where 
the  provision  of  a  sinking  fund  in  relation  to  an  old  loan  has 
been  entirely  overlooked. 

A  surplus  in  the  fund  may  arise  iu  several  ways ;  either  by 
an  increased  rate  of.  accumulation  or  by  the  payment  into  the 
fund  of  the  proceeds  of  sale  of  part  of  the  assets  representing 
the  security  for  the  loan,  or  a  realised  profit  upon  the  sale  of 
an  investment.  In  the  case  of  commercial  and  financial  under- 
takings, a  surplus  may  arise  upon  the  withdrawal  of  part  of  the 
loan  from  the  operation  of  the  fund.  Two  typical  examjjles  of 
this  nature  are  very  fully  discussed  in  Chapter  XVIII. 

In  all  such  problems  it  is  first  necessary  to  ascertain  the 
actual  position  of  the  fund  at  the  time  the  adjustment  is  required 
to  be  made,  and  this  may  be  expressed  in  terms  of  the  present 
investments  and  the  future  annual  increment  to  accrue  to  the 
fund.  The  problem  may  be  simplified  by  treating,  as  one 
factor,  the  "  Annual  Increment ''  of  the  fund  which  consists  of 
the  annual  instalment  and  the  income  to  be  received  from  the 
present  investments,  whether  the  rate  per  cent,  of  such  income 
is  the  same  as  the  rate  of  accumulation  or  is  different.  Any 
variation  in  such  rates  may  continue  during  the  whole  of  the 
unexpired  portion  of  the  repayment  period  or  for  a  portion  of 
the  period  only.  The  term  "  Annual  Increment "  is  fully 
discussed  in  Chapters  XIY.  and  XXII. 

The  principal  causes  giving  rise  to  a  necessity  to  make  an 
adjustment  of  the  fund  are  variations  in  the  rates  jDer  cent,  of 
accumulation  or  of  income  upon  the  present  investments  and 
variations  in  the  period  of  repayment,  or  a  combination  of  both 
rate  per  cent,  and  period.  Of  the  two  causes  a  variation  in 
either  of  the  rates  per  cent,  is  the  most  probable  for  many 
obvious  reasons,  and  it  is  very  important  that  this  should  be 
carefully  observed  and  immediately  corrected,  in  order  to  avoid 
the  necessity  at  some  future  time  of  having  to  make  a  substantial 
adjustment  due  to  the  accumulation  of  small  errors.  The 
longer  a  deficiency  is  allowed  to  accumulate  the  greater  becomes 
the  resulting  burden  imposed  upon  the  correspondingly  reduced 
number  of  the  final  years  of  the  redemption  period. 


8  REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

Calculation  of  a  Typical  Sinking  Fund.  Iu  order  to 
provide  an  example  wliicli  may  be  used  to  illustrate  the  wliole 
of  the  above  problems,  Chapter  XY,  Calculation  (XY)  1,  shows 
the  method  of  ascertaining-  the  annual  instalment  relating  to  a 
loan  of  £26,495  repayable  in  25  years  with  an  assumed  rate  of 
accumulation  of  3|  per  cent,  per  annum.  (xA^uthor's  standard 
calculation  form,  No.  3  x.) 

Methods  of  Adjustment.  Throughout  the  book  the  fact 
that  the  particular  method  adopted  is  not  the  most  direct  one 
has  been  left  entirely  out  of  consideration,  provided  it  has  an 
educational  value.  In  all  cases,  however,  the  shorter  and  more 
direct  method  has  been  shown  and  the  results  by  the  two 
methods  compared.  In  the  case  of  the  adjustment  due  to  a 
deficiency  in  the  fund,  in  Chapter  XY.,  four  methods  are  given, 
which  are  summarised  at  the  head  of  that  chapter.  The 
adjustment  of  a  deficiency  has  been  treated  in  this  exhaustive 
manner,  far  beyond  its  relative  importance,  in  order  to  present 
to  the  student  a  practical  example  illustrating  the  interdepend- 
ence of  the  present  value  and  future  amount  of  £1  and  of 
■£1  per  annum.  In  the  above  example,  as  well  as  in  later  ones,  a 
statement  has  been  prepared  showing  the  various  stages  by 
which  the  amended  annual  instalment  is  ascertained,  and  this  is 
followed  in  all  cases  by  a  further  statement  showing  the  final 
repayment  of  the  loan  by  the  operation  of  the  sinking  fund  and 
the  amended  annual  instalment  rendered  necessary  by  the 
variation  in  the  original  conditions. 

Wherever  required,  the  method  has  been  reduced  to  a  series 
of  stages  briefly  stated  giving  a  reference  to  the  individual 
calculations.  In  the  earlier  parts  of  the  book  the  actual  details 
of  the  calculations  are  given  in  full  or  in  the  Appendix,  but  in 
later  chapters  only  the  final  results  are  given  owing  to 
consideration  of  space  and  also  because  similar  examples  have 
previously  been  worked  out. 

The  Antjtual  Increment  Methods.  The  adjustments  next 
considered  are  those  due  to  a  variation  in  the  rate  per  cent, 
either  of  accumulation  or  income  from  investments,  a  variation 
in  the  period  of  repayment,  or  a  combination  of  the  two  factors 
of  rate  per  cent,  and  period.  As  in  the  case  of  a  deficiency  or  a 
surplus  in  the  fund,  the  amended  annual  instalment  is  first 
ascertained  hj  the  deductive  method  fully  described  in 
Chapter  XIX,  which  is  based  upon  the  consideration  of  the 
whole  of  the  factors  governing  the  fund.  The  same  result  is 
also  shown  by  the  annual  increment  (balance  of  loan)  method 


INTRODUCTION  9 

fully  described  in  Chapter  XXII.  In  the  case,  however,  of  a 
variation  in  the  rate  of  accumulation  accompanied  by  a 
variation  in  the  rate  of  income  from  investments  the  latter 
factor  is  eliminated  by  merging  it  in  the  annual  increment  and 
dealing  only  with  that  annual  sum.  The  varying  rate  of 
accumulation  then  becomes  the  only  outstanding  factor,  and  it 
is  therefore  possible  to  deduce  a  method  which  has  been  called 
"the  annual  increment  (ratio)  method,"  depending  upon  the 
ratio  existing  between  the  original  and  amended  rates  of 
accumulation.  The  whole  of  the  calculations  by  the  annual 
increment  (ratio)  method  relating  to  a  variation  in  the  rate  per 
cent,  only,  and  also  to  a  variation  of  the  period  of  repayment 
only,  bear  a  strong  family  likeness  and  are  capable  of  being 
reduced  to  simple  rules  and  formulse,  and  this  has  been  shown 
in  detail.  Having  in  Chapter  XXVI  discussed  a  combined 
variation  in  both  factors  of  rate  per  cent,  and  period  and  having 
again  deduced  a  formula  therefrom  the  whole  of  the  formulae 
so  obtained  have  been  reduced  to  simple  rules. 

The  Dates  of  Boreowing  and  Hepayment.  Up  to  this 
point  all  possible  causes  of  the  adjustment  of  a  sinking  fund 
have  been  exhausted,  but  the  subject  has  been  treated  from  the 
purely  mathematical  or  actuarial  standpoint,  namely,  that  all 
loans  are  borrowed  in  one  sum  at  the  beginning  of  the  financial 
year  and  that  the  annual  instalments  are  set  aside  at  the  end  of 
such  year.  The  actual  practical  conditions  are  next  dealt  with, 
namely,  that  the  loan  is,  as  a  rule,  borrowed  over  a  period  of 
years,  in  various  amounts  and  at  various  dates  in  any  year,  and 
is  repayable  sometimes  over  a  period  of  years,  but  often  on  a 
given  date.  The  subject  is  further  complicated  by  the  fact  that 
varying  periods  are  allowed  for  the  repayment  of  loans 
sanctioned  for  different  classes  of  outlay  depending  upon  the 
life,  or  duration  of  continuing  utility,  of  the  individual  works. 
And  this  varying  period  of  repayment  may  be,  and  often  is, 
complicated  by  practical  variations  in  the  dates  of  borrowing. 
This  part  of  the  subject  has  therefore  been  divided,  by  dealing 
first  with  loans  authorised  for  outlay  of  one  character  only 
where  the  problem  is  not  complicated  by  different  periods  of 
repayment  due  to  the  life  of  the  asset.  The  problems  relating 
to  the  dates  of  borrowing  are  sub-divided  as  follows  :  — 

(a)  Where  the  loan  is  borrowed  over  several  years,  in  one 
sum  in  each  year,  and  is  repayable  over  a  term  of  years 
in  a  prescribed  period  from  the  several  dates  of  borrowing. 

Chapter  XXVIII. 


lo  REPAYMENT   OF   LOCAL   AND    OTHER   LOANvS 

(6)  Where  the  loan  is  borrowed  over  several  years,  in  one 
sum  in  each  year,  and  is  repayable  in  one  sum  on  a 
certain  specified  date.  Chapter  XXIX. 

(c)  Where  the  loan  is  borrowed  in  one  or  more  years  in 
varying  amounts  and  at  varying  dates  in  each  year  and 
is  repayable  in  one  sum  on  a  certain  specified  date,  and 
it  is  further  required  that  the  revenue  or  rate  account 
of  each  year  of  borrowing  shall  be  charged  with  a 
proportionate  part  of  the  annual  sinking  fund  instalment. 

Chapter  XXX. 

In  the  case  of  loans  borrowed  over  a  series  of  years,  where 
the  repayment  is  spread  over  a  period  equal  to  the  extended 
years  of  borrowing,  the  amounts  borrowed  in  each  year  may  be 
treated  as  individual  loans,  and  the  only  points  to  be  considered 
are  administrative,  and  relate  to  the  number  of  sinking  funds, 
namely,  whether  it  is  preferable  to  keep  a  separate  sinking  fund 
for  each  year's  borrowings  or  to  keep  only  one  fund  for  the 
total  loan.  This  is  fully  discussed  in  Chapter  XXYIII,  and, 
as  there  stated,  cannot  be  applied  to  the  redemption  of  stock. 

In  the  case  of  loans  borrowed  over  a  period  of  years,  raised 
by  the  issue  of  stock  redeemable  on  a  fixed  date  the  several 
sinking  fund  instalments,  although  commencing  at  various 
dates,  yet  mature  on  the  same  date.  The  enquiry  is  still 
confined  to  loans  in  respect  of  outlay  of  one  nature  and  having 
a  uniform  period  of  repayment.  This  class  has  been  sub- 
divided into  two  groups,  and  is  fully  discussed  in  Chapter 
XXIX. 

1.  Where  the  date  of  repayment  is  known  at  the  time  the 

money  is  borrowed. 

2.  Where  the  date  of  repayment  is  fixed  after  the  sinking 

fund  has  been  in  operation  for  a  number  of  years,  and 

an  adjustment  becomes  necessarj'. 
The  apportionment  of  a  part  of  a  full  year's  instalment  to 
be  charged  against  the  revenue  or  rate  account  of  the  year  in 
which  the  money  is  borrowed  is  treated  fully  in  Chapter  XXX. 
As  a  rule,  this  may  be,  and  generally  is,  ignored;  but  the 
particular  circumstances  in  connection  with  a  large  loan  may 
render  it  advisable  to  make  a  charge  of  this  nature.  There  are 
several  interesting  features  in  the  method  which  is  illustrated  by 
the  example  in  Chapter  XXX,  and  which  may  be  compared 
with  the  instalments  to  be  set  aside  when  the  year  of  borrowing 
is  not  charged  with  any  such  proportional  annual  instalment, 
as  in  Chapter  XXIX.     Stated  briefly,  the  effect  is  to  ante-date 


INTRODUCTION  ii 

the  charge  to  revenue  or  rate  and  to  impose  an  increased 
burden  upon  the  years  of  borrowing.  The  third  year  of  the 
sinking  fund  period  is  charged  with  the  same  amount  under 
each  method  because  the  repayment  period  is  assumed  to 
commence  at  the  conclusion  of  the  first  year  of  borrowing. 
Such  increased  annual  burden  during  the  earlier  years  operates 
by  way  of  relief  to  the  remainder  of  the  repayment  period, 
but  only  to  a  slight  extent.  This  problem  furnishes  another 
example  of  an  adjustment  being  required  in  consequence  of 
irregular  contributions  to  the  fund  during  the  earlier  years. 

The  Life  of  the  Asset,  and  the  Equation  of  the  Period 
OF  Repayment.  Having  dealt  with  problems  relating  solely  to 
the  adjustment  of  the  sinking  fund,  owing  to  causes  of  a  purely 
actuarial  or  mathematical  nature,  there  is  still  to  be  considered 
the  more  difficult  subject  of  the  variation  in  the  periods  allowed 
for  the  redemption  of  loans  for  large  public  works,  where  each 
component  part  of  the  outlay  has  a  different  life  or  duration  of 
continuing  utility.  This  variation  in  the  redemption  period 
has  not  any  disturbing  effect  when  the  loan  is  authorised  for 
one  class  of  outlay  only,  or  is  in  respect  of  several  classes  of 
outlay,  each  having  the  same  period  of  repayment.  But  loans 
are  now  often  authorised  for  large  public  works  which  include 
various  classes  of  outlay,  each  class  having  its  oAvn  redemption 
period,  based  upon  its  duration  of  continuing  utility,  and  also 
forming  a  variable  proportion  of  the  total  cost,  and  it  is 
required  that  the  total  loan  shall  be  repaid  on  the  same  date. 
In  such  cases  it  becomes  necessary  to  ascertain  the  equated 
period  of  repayment.  The  same  necessity  arises  on  the  consoli- 
dation of  existing  loans  repayable  at  various  future  dates,  but 
in  such  cases  the  problem  is  further  complicated  by  the  amounts 
then  standing  to  the  credit  of  the  individual  sinking  funds, 
the  value  of  the  investments  representing  each  fund,  the  rate  of 
income  arising  therefrom,  and  also  the  incidence  of  the  present 
redemption  charges  upon  different  departments  of  the  local 
authority.  It  may  be  stated  generally  that  the  problem  of  the 
equation  of  the  period  of  repayment  applies  equally  to  all  such 
cases  and  that  in  fixing  the  equated  date  of  repayment  there  are 
two  interests  to  be  considered,  namelj^,  the  loanholder,  who  looks 
only  for  the  due  payment  of  his  principal,  and  the  annual 
interest  thereon,  and  the  individual  ratepayer  who  is  required 
to  provide  his  proper  portion  of  the  annual  amount  which 
Parliament,  or  the  government  department  concerned,  has  laid 
down  in  principle  as  the  annual  wastage  of  the  assets  created 


12     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

out  of  the  loan.  This  annual  wastage  of  the  asset  is  imported 
into  the  problem  owing  to  the  fact  that  the  period  allowed  for 
the  repayment  of  the  loan  is  based  upon  the  life,  or  duration  of 
continuing  utility,  of  the  asset.  As  regards  the  loauholder,  the 
problem  is  a  simple  one.  In  all  cases  he  will  be  repaid,  at  some 
future  date,  the  amount  which  he  originally  advanced  to  the 
local  authority,  and,  in  addition,  he  will  receive  until  the 
repayment  of  the  loan,  interest  at  the  rate  per  cent,  originally 
fixed.  The  only  question  remaining  therefore,  so  far  as  he  is 
concerned,  is  the  relation  between  the  rate  of  interest  agreed  to 
be  paid  by  the  local  authority  and  the  rate  per  cent,  obtainable 
upon  the  open  market  when  the  loan  is  proposed  to  be  repaid. 
This  arises  only  upon  the  consolidation  of  existing  loans  repay- 
able at  fixed  future  dates  where  the  effect  of  consolidation  is  to 
vary,  and  generally  to  anticipate,  the  date  at  which  the  loan 
was  originally  repayable.  As  regards  the  loanholder,  therefore, 
the  most  important  factor  is  the  period  during  which  he  will 
continue  to  receive  interest  upon  the  loan  at  the  present  rate 
payable  by  the  local  authority.  If  the  present  rate  so  payable 
be  a  high  one  and  the  current  rate,  to  be  obtained  upon  the  open 
market,  at  the  time  when  the  local  authority  propose  to  repay 
the  loan,  be  expected  to  be  lower,  the  loanholder  will  naturally 
object  to  any  variation  of  the-  original  conditions,  and,  per 
contra,  he  will  gladly  accept  an  earlier  repayment  of  the  loan 
if  thereby  he  may  expect  to  obtain  a  higher  rate  of  interest 
upon  his  investment.  Consequently  the  loanholder  must  be 
consulted,  and  his  consent  obtained,  before  any  change  be  made 
in  the  original  conditions  upon  the  consolidation  of  loans.  This 
uncertainty  as  to  the  future  rate  of  interest  is  one  of  the  reasons 
why  Parliament  has,  for  some  years  past,  refused  to  sanction 
the  issue  of  an  irredeemable  stock  in  consequence  of  the  difficulty 
in  applying  the  amount  in  the  sinking  fund  to  its  proper 
purpose.  The  stock  can  thus  only  be  redeemed  by  purchase 
upon  the  open  market,  and  the  premium  paid  upon  such 
occasions  cannot  be  taken  out  of  the  sinking  fund,  but  must 
be  charged  against  the  revenue  or  rate  account  of  the  current 
year. 

The  ratepayer,  on  the  contrary,  is  in  a  very  different 
position,  in  that  the  money  paid  to  the  loanholder  by  way  of 
interest  \\\wu  tlie  loan,  and  the  annual  sums  set  aside  out  of 
revenue  or  rate  to  redeem  the  debt  are  paid  by  him.  But  the 
ratepayer  comes  and  goes,  whilst  the  loanholder  goes  on  for 
ever,  or  at  least  until  his  loan  is  repaid.  The  loanholder 
naturallv  cinisidcrs  the  value  of  his  investment  and  tlie  interest 


INTRODUCTION  13 

to  be  derived  therefrom,  and  the  state  of  the  money  market  both 
at  the  present  time  and  in  the  future  are  to  him  very  important 
factors.  The  ratepayer,  on  the  contrary,  considers  only  the 
annual  amount  paid  by  him  by  way  of  rate,  and  compound 
interest  is  to  him  a  negligible,  if  not  an  unknown  term.  In 
addition,  he  is  never  consulted  individually  as  to  the  annual 
amount  of  rate  which  he  may  be  called  upon  to  pay.  He  may 
be  invited  to  attend  a  meeting  called  to  approve  or  disapprove 
of  a  Bill  to  be  laid  before  Parliament  to  authorise  the  spending 
of  money  on  capital  account,  but  he  is  generally  ignorant  of 
the  matter,  and  is  too  busy  trying  to  earn  the  amount  he  has 
to  pay  by  way  of  rate,  to  attend  any  such  meetings.  The  result 
is  that  the  final  adjustment  is  left  entirely  to  the  officials  of  the 
local  authority  subject  only  to  the  control  of  Parliament  or  the 
Local  Government  Board,  and  the  next  .step  therefore  is  to 
investigate  the  methods  generally  adopted  in  order  to  arrive  at 
the  equated  period  of  repayment  and  the  consequent  amended 
annual  sinking  fund  instalment  to  be  charged  to  revenue  or 
rate. 

Before  doing  so,  however,  it  should  be  pointed  out  that 
any  necessity  to  fix  the  equated  period  did  not  arise  to  any 
great  extent  until  it  became  the  common  practice  of  local 
authorities  to  issue  stock  or  to  consolidate  existing  loans  re- 
payable at  various  dates.  Prior  to  that  time,  any  variation 
in  the  periods  of  repayment  alloAved  for  different  classes  of 
outlay  was  met  by  keeping  separate  funds  for  each  amount  of 
loan  having  the  same  repayment  period  and  allowing  each  fund 
to' mature  at  the  due  date.  The  relation  between  the  life  of 
the  asset  and  the  consequent  annual  loan  charge  upon  the 
revenue  or  rate  accounts  of  successive  years  is  fnlly  discussed  in 
Chapter  XXXII,  where  it  is  found  that  the  variation  in  the 
periods  of  repayment  allowed  is  not  of  itself  a  cause  of  an 
equation  being  required,  which  depends  upon  a  combination  of 
two  factors,  namely,  the  variable  period  of  repayment  and  the 
obligation  to  repay  A^arious  loans  on  one  instead  of  on  different 
dates.  The  problem  arising  on  the  consolidation  of  stocks  or 
loans  repayable  at  various  dates  is  exactly  similar  in  principle 
althouo-h  arising  in  a  somewhat  different  manner,  but  is  further 
complicated  by  the  amount  in  the  fund  at  the  time  of  making 
the  adjustment. 

The  Equation  of  the  Peeiod  of  Repayment.  The 
equation  of  the  period  of  repayment  has  been  considered  from 
two   points   of  view,    namely,   one   relating   to   the   method   of 


14     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

ascertaining  the  equated  date  and  tlie  other  to  the  incidence  of 
the  annual  burden  upon  the  revenue  or  rate  account.  These 
two  points  are  fully  treated  in  Chapters  XXXII,  XXXIII  and 
XXXIV.  The  method  generally  adopted  to  find  the  equated 
period  is  the  arithmetical  one  known  as  the  "  equation  of  pay- 
ments, ■"  which  is  fully  described  in  Chapter  XXXII.  It  is 
there  proved,  by  two  examples  worked  out  in  detail,  that  the 
equated  period  as  generally  adopted  is  not  the  true  equated 
period  and  that  the  effect  of  adopting  it  is  to  extend  the  period 
of  repayment  beyond  the  true  or  mathematically  equated  period. 
This  may  not  be  important  in  many  cases,  but  may  be  extremely 
so  in  the  case  of  very  large  loans ;  and,  if  it  be  necessary  to  make 
such  an  adjustment  at  all,  it  is  surely  imperative  that  the 
principle  upon  which  it  is  made  is  scientifically  accurate. 

Having  described  the  proper  method  of  finding  the  true 
equated  period,  Chapter  XXXIII  is  occupied  with  an 
examination,  illustrated  by  the  actual  example  used  in  Chapter 
XXXII,  of  the  effect  of  the  generally  adopted  practice  of  fixing 
the  amended  annual  sinking  fund  instalment  by  spreading  the 
burden  equally  over  the  whole  of  the  equated  period  as  if  it 
related  to  an  original  loan,  repayable  in  such  period,  the  whole 
of  the  loan  representing  outlay  of  one  character  only,  having  a 
life  or  period  of  utility  of  that  length.  The  method  is  a  simple 
one,  but  is  wrong  in  principle  although  it  has  received  the 
approval  of  many  years'  adoption. 

If  the  preliminary  stages  in  the  sanction  of  a  loan  be  care- 
fully reviewed  it  will  be  recognised  that  much  thought  and  care 
are  expended  in  determining  the  proper  periods  to  be  allowed 
for  the  repayment  of  loans  authorised  for  different  classes  of 
outlay.  The  whole  question  is  still  in  a  transition  state,  and 
great  divergence  exists  in  the  conditions  now  in  force ;  the  only 
recognised  factors  being  that  in  future  the  annual  charges  for 
redemption  of  the  debt  shall  bear  a  definite  relation  to  the  life 
of  the  asset,  with  a  further  extension  of  the  principle,  that  even 
in  the  case  of  works  of  almost  permanent  utility  the  repayment 
shall  not  extend  beyond  a  certain  number  of  years.  This  latter 
requirement  is  to  protect  the  interests  of  future  generations  of 
ratepayers.  The  relation  between  the  period  of  repaynu'ut  and 
the  life  or  duration  of  continuing  utility  of  the  asset  is  imposed 
in  order  to  ensure  that  the  present  generation  shall  contribute, 
year  by  year,  the  proper  portion  of  the  wastage  of  the  asset.  In 
the  case  of  a  loan  raised  for  works  comprising  outlays  of  varying 
nature  with  varying  periods  of  repayment  and  where  separate 
sinking  funds  are  kept  in  respect  of  each  class  of  outlay  the 


INTRODUCTION  15 

principle  is  carried  out  exactly  because  the  earlier  years  bear 
the  heaviest  burden,  as  they  should  do,  owing  to  the  fact  that 
classes  of  outlay  having  a  short  life  Avill  be  worn  out  and 
require  replacing  at  the  end  of  the  period.  Under  these  condi- 
tions the  loan  is  entirely  repaid  by  the  time  the  works  cease  to 
be  of  utility  or  are  worn  out. 

The   Equation   of  the   Peeiod   of   Repayment,    and   the 
Incidence  of  Taxation.     The  Annual  Instalment.       Under 
the  present  practice  on  equation  the  above  principle  is  departed 
from,  and  the  burden  is  spread  equally  over  the  equated  period 
with  a  total  disregard  to  any  period  of  utility,  and  as  demon- 
strated in  Chapter  XXXIII,  there  is  actually  considerable  relief 
to  the  early  years  of  the  equated  period  as  well  as  an  entire 
removal  of  any  burden  during  the  years  of  the  original  period 
beyond  the  equated  period.     As  a  consequence,   the  whole  of 
this  relief  is  imposed  as  an  additional  burden  of  considerable 
magnitude  upon  the  final  years  of  the  equated  period.     There 
is  here  a  total  reversal  of  the  generally  accepted  principle  of 
spreading  the  repayment  of  the  loan  over  the  period  represented 
by  the  life  of  the  asset,  accompanied  by  an  absolute  injustice  to 
a  section  of  the  ratepayers.     By  adopting  the  equated  method 
in  general  use,  as  applied  to  works  consisting  of  various  classes 
of  outlay,  and  also  on  the  consolidation  of  loans,  the  present 
generation  relieve  themselves  of  a  liability  to  contribute  their 
fair  share  of  the  burden  which  has  been  fixed  after  careful 
enquiry  by  Parliament;  and  thereby  impose  an  extra  burden 
upon  future  years.     In  addition  they  also  postpone  the  repay- 
ment of  the  loans  with  shorter  periods  which  would  have  been 
repaid  during  the  earlier  years,  and  the  result  is  that  money 
cannot  properly  be  reborrowed  to  replace  assets  with  a  shorter 
life  than  the  equated  period  because,  when  they  are  worn  out, 
the  original  loan  has  not  been  repaid  by  means  of  the  sinking 
fund.     A  remedy  for  this  state  of  affairs,  so  far  as  the  annual 
instalment  only  is  concerned,  is  pointed  out  in  Chapter  XXXIV, 
namely,  by  spreading  the  burden  over  the  equated  period,  not 
by  an  equal  annual  instalment,  as  is  the  present  practice,  but 
by   instalments    of   varying    amounts   approximating   to   those 
originally  imposed  which  were  based  upon  the  life  of  the  asset. 
The  principle  of  this  method  is  to  ascertain,  first,  the  amount 
of  loan  which  will  be  provided  at  the  end  of  the  equated  period, 
by  the  accumulation  of  the  annual  instalments  as  originally 
fixed.     The    amount    of    such    instalments    at    the  end    of   the 
number  of  years  for  which  they  would  have  been  set  aside  under 


i6  REPAYMENT    OF   LOCAL   AND    OTHER   LOANvS 

the  original  conditions  slionld  be  ascertained,  and  if  any  of 
these  periods  are  shorter  than  the  equated  period,  the  amount 
at  the  end  of  such  periods  should  be  further  accumulated  until 
the  end  of  the  equated  period.  The  difference  between  the 
amount  of  loan  so  ascertained  and  the  total  amount  of  the  loan 
ultimately  repayable  will  represent  the  amount  to  be  provided 
by  supplementary  annual  instalments  to  be  spread  over  the 
equated  period,  due  to  the  fact  that  the  equated  period  is 
shorter  than  the  original  periods  allowed  for  the  repayment  of 
the  parts  of  the  loan  with  longer  periods,  and  that  the  relief 
afforded  by  the  equation  to  these  later  years  should  be  borne 
equitably  by  each  year  of  the  equated  period.  Strictly 
speaking,  such  supplementary  annual  instalments  should  be 
graded  in  some  manner  proportionate  to  the  original  annual 
instalments  which  were  based  upon  the  life  of  the  asset,  and 
although  the  calculation  is  fullv  described  it  is  somewhat 
intricate,  and  the  justice  of  the  case  will  generally  be  met  by 
spreading  this  supplementary  annual  instalment  equally  over 
the  equated  period. 

The  Equation  of  the  Peeiod  of  Repayment,  and  the 
Incidence  of  Taxation.  Interest  upon  the  Loan.  The  result 
of  spreading  the  annual  instalment  over  the  equated  period  in 
proportion  to  the  instalments  before  equation  is  shown  in  Table 
XXXIA',  J,  where  the  original  annual  instalments  are  corrected 
in  this  manner.  On  referring  to  Table  XXXIII,  B,  it  Avill  be 
seen  that  the  effect  of  equating  the  period  is  to  throw  a  heavy 
additional  burden  upon  the  final  year's  of  the  equated  period 
in  respect  of  interest  upon  the  loan.  In  Chapter  XXXIY, 
a  method  is  described  of  distributing  the  redemption  charge 
(the  annual  instalment)  equitably  over  the  equated  period,  and 
in  Chapter  XXXV  a  similar  course  is  adopted  with  regard  to 
the  interest  upon  the  loan,  with  the  result  shown  in  Table 
XXXY,  C.  By  combining  the  correctly  equated  annual  instal- 
ments shown  in  Table  XXXIV,  J,  with  the  correctly  equated 
annual  interest  charges  shown  in  Table  XXXV,  C,  the  total 
annual  loan  charges  during  the  equated  period  may  be  ascer- 
tained as  shown  in  Table  XXXV,  F.  The  subject  is  so 
important  that  the  result  has  been  shown  in  graphic  form, 
which  is  fully  described  and  explained  in  Chapter  XXXV. 
In  order  to  express  in  actual  values  the  effect  of  the  above 
adjustment  both  as  to  the  instalment  and  interest  upon  the 
loan  it  may  be  stated  in  terms  of  annual  rate.  It  has  been 
"•iven  in  evidence  before  a  Parliamentarv  Committee  that  in  one 


INTRODUCTION  17 

particular  case  of  consolidation  of  loans  the  immediate  effect  of 
an  equation  of  the  period  of  repayment  was  a  saving  of  three- 
pence in  the  £  in  the  annual  rate.  In  this  connection  it  should 
be  remembered  that  as  shown  in  Table  XXXIII,  C,  there  is 
not  any  difference  in  the  annual  charges  for  interest  upon  the 
loan  during  the  early  years  of  the  equated  period,  before  and 
after  equation,  but  that  the  decrease  is  entirely  in  the  annual 
instalment.  Adopting  the  figure  of  threepence  in  the  pound  as 
a  standard,  the  result  in  the  present  case  would  be  as  follows, 
including  the  interest  upon  the  loan  as  well  as  the  annual 
instalment.  The  following  figures  may  be  converted  into 
American  currency  by  adopting  the  equivalent  of  2^  cents  to 
the  dollar  instead  of  6d.  in  the  pound  :  — 

Decreased  rate  Increased  rate 

Original  redemption  period.  per  £.  of  per  £  of 

Annual  Value.         Annual  Value. 

(  5  years     ...     652  pence 
Equated  period  ...-10  years     ...     3-20  pence 

(  8  years     ...  ...  11  "85  pence. 

Post-equated  period  ...        6  years     ...     9'76  pence 

16  years     ...     2  79  pence 

Actual  Calculations.  The  method  adopted  throughout 
the  book  is  to  insist  upon  a  very  careful  scrutiny  of  the  present 
and  future  conditions  and  also  of  the  actuarial  and  mathe- 
matical principles  involved.  It  is  very  important  to  prove  all 
ascertained  results,  both  as  to  method  and  accuracy  of  computa- 
tion, seeing  that  the  actual  working  out  of  the  fund  will  occupy 
many  years  and  the  effect  of  any  present  error  will  be  serious 
if  it  be  allowed  to  accumulate  for  any  length  of  time.  Mere 
repetition  of  the  actual  calculation  is  not  sufficient.  A  far 
preferable  method  is  to  work  out  the  operation  of  the  fund 
year  by  year  by  the  arithmetical  method  as  shown  in  the  pro 
forma  accounts  already  referred  to.  This  should  be  done  in  all 
cases  without  exception  before  the  problem  is  finally  disposed 
of,  but  the  method  is  laborious  and  much  time  is  wasted  if  the 
original  calculation  be  wrong.  The  best  way  is  to  prove  the 
result  by  mathematical  means  which  are  much  shorter,  either 
by  adopting  an  alternative  method,  of  which  many  instances 
are  given  in  the  various  chapters,  or  by  comparing  the  amended 
with  the  original  annual  instalment  and  accounting  for  the 
difference.  The  actual  arithmetical  calculation  of  the  pro 
forma  sinking-  fund  accounts  mav  be  left  to  a  iunior  official, 
but  it  will  save  him  considerable  time  if  the  senior  first  ascer- 


i8  REPAYMENT   OF    LOCAL    AND    OTHER   LOANS 

tains  the  amount  in  the  fund  at  the  end  of  every  five  or  ten 
years  by  means  of  the  tables  or  otherwise  as  described  in  the 
various  chapters.  The  whole  of  the  calculations  in  the  book 
have  been  verified  in  this  manner  and  in  many  cases  the  method 
of  proof  is  shown  in  detail.  In  cases,  however,  where  it  is  not 
shown  the  verification  has  been  made  and  is  only  omitted  for 
want  of  space. 


Section  I. 
Mathematical   Principles. 


CHAPTER  II. 
LOGARITHMS. 

Advantage  of  use  of  logs.     History.     Connection  between 

logs.  and  arithmetical  and  geometrical  progressions. 

Definition.     Various  arithmetical  calculations  by  logs. 

Logs,    of    numbers    between    even    multiples    of     10. 

Characteristic.     Mantissa.     Method  of  dividing  a  log. 

with  a  negative  characteristic.   Method  of  dividing  one 

log.  by  another. 
In  a  work  of  this  nature,  dealing  with  calculations  which 
are  based  upon  the  higher  branches  of  mathematics,  it  is 
obvious  that  the  ordinary  methods  of  arithmetic  are  inadequate, 
and  that  the  aid  of  logarithms  must  be  invoked  even  if  the 
fullest  use  be  made  of  the  various  published  tables  of  compound 
interest.  There  is  a  limit  to  the  number  of  rates  per  cent. 
which  may  be  included  in  any  table,  and  it  is  often  required  to 
make  a  calculation  at  a  rate  per  cent,  not  worked  out.  In  such 
cases  it  is  necessary  to  revert  to  the  original  formulae,  all  of 
which  involve  raising  numbers,  containing  as  many  as  five  or 
six  figures,  to  the  power  of  the  number  of  years,  and  the  method 
of  continued  multiplication  becomes  too  laborious  and  uncertain. 
Even  when  using  the  tables  it  is  always  necessary  to  multiply  or 
divide  by  the  numbers  (containing  five  or  six  figures)  given  in 
the  tables,  and  a  great  saving  of  time  and  labour  is  effected  by 
doing  this  by  the  aid  of  logarithms.  But  beyond  the  little  time 
expended  in  becoming  familiar  with  the  method  of  using  such  a 
table,  there  is  not  any  greater  difficulty  than  in  using  any 
ordinary  commercial  ready  reckoner. 

This  chapter  deals  only  generally  with  the  subject  of 
logarithms,  and  as  the  use  of  this  book  cannot  be  complete 
without  a  copy  of  Inwood's  or  other  similar  tables,  and  a 
reliable  table  of  logs.,  for  a  fuller  acquaintance,  reference  must 
be  made  to  the  introductory  chapter  which  will  be  found  in  most 
log.  tables,  or  else  to  some  good  advanced  arithmetic. 

Logarithms  were  invented  by  John  Napier,  Baron  of 
MerchivSton,  in  Scotland,  who  published  his  first  work  in 
Edinburgh  in  1614.  This  work  contained  only  the  logarithms 
of  natural  sines,  and  are  not  what  are  now  known  as  Naperian 
or  hyperbolic  logarithms,  which  are  used  in  mathematical 
investigations  only  and  are  not  the  logarithms  in  common  use 


22  REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

to-day.  jS^apier  died  in  IGIT,  and  a  further  work  by  bim, 
edited  by  bis  son,  was  published  in  1619, 

The  first  published  table  of  decimal  or  common  logarithms 
was  published  by  Henry  Briggs,  Professor  of  Geometry  at 
Gresham  College,  London,  and  afterwards  Savilian  Professor 
of  Geometry  at  Oxford,  Avho  visited  jN"apier  in  1615.  Briggs 
published  his  table  in  1617  (after  the  death  of  Napier),  and 
these  logarithms  which  are  in  common  use  to-day  are  calculated 
to  a  base  of  10.  Briggs'  first  tables  contained  only  the 
logarithms  of  numbers  from  unity  to  1,000  to  14  places  of 
decimals.  The  arithmetical  calculation  of  logarithms  as  used 
by  Briggs,  is  a  very  laborious  process,  and  it  was  not  until 
1628  that  the  table  was  extended  for  numbers  from  unity  to 
101,000  by  Briggs  and  Adrian  Vlacq,  of  Gouda,  in  Holland. 
But  the  arithmetical  method  of  Briggs  Avas  afterwards  super- 
seded by  shorter  methods  depending  upon  more  advanced 
mathematical  rules,  and  there  are  now,  as  the  result  of  all  this 
labour,  very  accurate  tables  which  are  in  universal  use,  and  by 
means  of  which  very  intricate  calculations  may  be  made  by 
very  simple  methods. 

The  principle  upon  Avhicli  a  logarithm  is  based  is  exceedingly 
simple,  and  is  founded  upon  the  relation  existing  between  an 
arithmetical  and  a  geometrical  progression.  An  arithmetical 
progression  is  a  series  of  numbers  each  of  which  is  found  by 
adding  a  constant  number  to  the  previous  term  in  the  series,  the 
constant  number  so  added  being  called  the  ratio.  A  geometrical 
progression  is  a  series  of  numbers  each  of  which  is  found  by 
multiplying  the  previous  term  in  the  series  by  a  constant 
number,  such  constant  multiplier  being  called  the  ratio. 
Taking  a  series  of  numbers  in  geometrical  progression,  with  a 
ratio  of  10,  which  is  the  one  adopted  in  the  Briggean  or  decimal 
or  common  logarithms,  and  commencing  the  series  with  unity, 
the  following  series  is  obtained  :  — 

Geometrical 
Progression,      1.  10.  100.  1,000.  10,000.  100,000.  1,000,000. 

Taking  another  series  of  numbers  in  arithmetical  progression, 
witli  a  ratio  of  1,  and  commencing  the  series  with  0,  the 
f ollow^ing  series  is  obtained  :  — 

Arithmefical 
Pro;jr('s.siui,,     0.      1.        2.  3.  4.  5.  6. 

The  above  geometrical  progression  will  now  be  re-written 
expressing  eacli  term  by  the  power  of  10  wliich  it  represents, 
and  under  it  the  above  arithmetical  progression,  as  follows  :  — 


LOGARITHMS  23 

Geoinctricnl 
Progression,  10°  10^     10=         10^  10*  10^  10^ 

Arithmetical 
Progression,     0       12  3  4  5  6 

It  will  be  at  once  noticed  that  the  index  of  each  term  in  the 
geometrical  progression  is  the  same  as  the  corresponding  term 
in  the  arithmetical  progression. 

If  it  be  assumed  that  the  terms  in  the  arithmetical  series 
are  the  logarithms  of  the  corresponding  terms  in  the  first 
geometrical  series,  this  is  exactly  what  Briggs  did  when  he 
adopted  10  as  the  basis  of  his  system,  as  follows  :  — 


The  log.  of                  1. 

= 

10° 

= 

0 

10. 

= 

101 

= 

1 

100. 

= 

102 

= 

0 

'V 

1,000. 

= 

103 

= 

a 

10,000. 

= 

104 

= 

4 

100,000. 

= 

105 

= 

5 

1,000,000. 

= 

106 

= 

6, 

and  so  on, 

and  therefrom  the  definition  of  a  common  logarithm  may  be 
expressed,  viz.,  the  logarithm  of  a  number  [to  tJic  base  10)  is 
the  poiver  or  index  to  ivhich  10  has  to  be  raised  to  yroduce  that 
number,  for  example  :  the  logarithm  of  10,000,  to  the  base  10, 
is  the  power  or  index  4,  to  which  10  has  to  be  raised  to  produce 
10,000,  but  as  10  is  the  common  base  to  which  all  numbers 
are  reduced,  the  indices,  or  logarithms,  only  are  required,  and 
the  decimal  part  of  this  index  is  all  that  is  given  in  the  log. 
tables. 

The  logarithms  of  numbers  which  are  even  powers  of  10  may 
be  ascertained  in  the  above  simple  manner,  and  attention  has 
been  drawn  to  the  enormous  labour  involved  in  calculating  the 
logarithms  of  the  intermediate  numbers.  It  is  not  necessary  to 
enquire  deeper  into  the  methods,  but  only  to  accept  the  tables 
which  are  the  product  of  that  labour.  The  logs,  which  have 
been  already  found  may  be  used  to  illustrate  the  various 
advantages  of  their  use,  taking  familiar  arithmetical  calcula- 
tions, using  the  actual  numbers  in  the  ordinary  way,  and  then 
repeating  the  calculations,  using  the  logs,  of  the  numbers 
instead  of  the  actual  numbers,  as  follows  :  — 

Miiltiplieatiori : 

10x1,000       =       10,000.  by  logs.,  1  +  3  =  4,  or  log.       10,000 
Division : 

100,000-100=         1,000.  by  logs.,  5-2  =  3,  or  log.  1,000 


24     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Involution : 

1003  =1,000,000.  by  logs.,  2x3  =  6,  or  log.  1,000,000 

Evolution  : 


V  1,000,000   =  100.  by  logs.,  6-3  =  2,  or  log.  100 

It  will  be  seen  therefore  that  by  the  use  of  logs. — 

Multiplication'!  /       addition       \ 

Division  |  of  ordinary  \  subtraction    I 

T        1    ,•  /  1       "   -becomes-,        -,..   -,.     ,.      (respective 

Involution  numbers   j  muitipiication         ^ 

Evolution         /  I       division        '  ^ 

It  is  a  common  practice  when  multiplying  together  two 
powers  of  10,  such  as  (10  x  1,000)  to  write  down  1,  and  add 
4  cyphers,  thus,  10,000,  being  the  sum  of  the  cyphers  in  the 
two  numbers.  This  is  actually  a  logarithmic  method  of  calcula- 
tion often  used  by  people  who  do  not  know  anything  about 
logarithms.  This  is  the  principle  of  the  algebraical  theory  of 
indices,  of  which  the  following  is  an  illustration  :  — 

Multiplication  :  x  x  x^  =  x'^i+S)  =  .x* 

Division  :  x^-^x"  =  .x*^" -*  =  x^ 

Involution  :  {x'^  Y  =  x^^^^^  =  x^ 

Evolution:  jx^    =x^''^^^  =  x" 

The  above  operations  correspond  to  the  previous  illustrations 
in  which  the  actual  powers  of  10  were  used.  In  the  algebraical 
form  above,  10  has  been  replaced  by  a:,  with  the  result  that 
similar  logs,  are  obtained  in  each  set  of  examples. 

Up  to  this  point  only  whole  numbers  have  been  considered 
which  are  even  multiples  of  10,  and  of  which  the  logs,  are 
whole  numbers  above  unity,  and  it  has  been  ascertained  that  the 
logarithm  of  10  is  1,  that  of  100  is  2,  and  so  on  for  any  even 
power  of  10.  It  is  therefore  obvious  that  the  logs,  of  numbers 
less  than  10  must  be  fractions.  This  also  applies  to  numbers 
between  10  and  100,  the  logs,  of  which  must  be  between  1  and  2, 
and  equally  to  numbers  between  any  two  consecutive  powers  of 
10,  which  logs,  consist  of  a  whole  number  and  a  fractional  part, 
the  whole  number  being  the  log.  of  the  next  loAver  power  of  10. 

A  logarithm  then  consists  of  two  parts — the  integral  part, 
which  is  called  the  Characteristic,  and  the  fractional  or  decimal 
part,  which  is  called  the  Mantissa,  and  all  logs,  are  expressed 
in  decimals,  usually  to  7  places. 

The  Mantissa  (or  fractional  part)  is  always  positive,  and  is 


LOGARITHMS  25 

always  tlie  same  for  any  one  combination  of  figures,  irrespective 
of  the  place  of  the  decimal  point. 

The  Characteristic  represents  merely  the  position  of  the 
decimal  point  in  the  number  which  the  log.  represents,  and 
changes  only  after  passing  each  power  of  10.  The  characteristic 
is  in  all  cases  the  power  to  which  10  has  to  be  raised  to  produce 
the  next  lower  power  of  10. 

The  logarithm  tables  contain  only  the  mantissa  part  of  the 
logarithm  corresponding  to  the  particular  combination  of  figures 
forming  the  number  of  which  the  log.  is  required,  and  since 
these  figures  may,  according  to  the  position  of  the  decimal 
point,  represent  either  whole  numbers  or  fractions,  the 
characteristic  is  positive  in  the  case  of  a  whole  number,  and 
negative  in  the  case  of  a  fractional  number. 

It  will  be  noticed,  on  referring  to  the  logs,  of  the  powers 
of  10,  referred  to  above,  that  the  log.  of  1,000,000  is  6,  or  one 
less  than  the  number  of  integral  figures  (seven)  in  the  number, 
and  similarly  with  the  other  powers  of  10,  and  the  rule  applies 
generally,  as  will  be  seen  by  taking  the  logarithm  of  the 
number  26495. 

In  the  table  of  logs,  opposite  26495  are  the  figures  423,1639 
which  is  the  mantissa  of  the  log.  of  any  number  containing 
the  figures  26495  in  this  order,  whether  preceded  or  followed 
by  any  number  of  cyphers.  The  actual  position  of  the  decimal 
point  determines  the  characteristic  or  integral  part  of  the  log. 
as  follows  :  — 

Log.  26495  =4-423  1639 

2649-5  =3-423  1639 

264-95  =2-423  1639 

26-495  =1-423  1639 


2-6495     =0-423  1639 


•26495  =1-423  1639 
•026495  =2-423  1639 
•0026495  =  3-423  1639,  and  so  on. 

On  comparing  the  above  logs,  with  the  logs,  of  the  powers 
of  10  previously  given,  it  will  be  noticed,  for  instance,  that 
264.95  (being  above  100  and  below  1,000)  has  the  characteristic 


26  REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

2  as  previously  explained,  and  it  will  be  further  noticed  that 
as  the  decimal  point  in  the  original  number  is  moved  place  by 
place  to  the  left  (equivalent  to  dividing  the  previous  number 
by  10)  the  characteristic  of  the  logarithm  is  reduced  by  1. 
But  in  the  case  of  2'6-l:95  the  characteristic  becomes  0,  and  as 
the  number  is  further  divided  by  10  and  the  decimal  point 
moved  still  further  to  the  left,  it  becomes  -1,  -2,  —3,  and  so 
on.  The  characteristic  being  the  only  negative  part  of  the  log., 
the  minus  sign  is  placed  over  it  instead  of  to  the  left. 

A  fflance  at  the  above  lof>-s.  will  show  that  the  characteristic 
follows  two  rules,  viz. :  — 

(1)  In  the  case  of  nuiiihers  yreater  than  unity,  the  charac- 
teristic is  one  les.s  titan  the  7iiimher  of  integral  figures 
in  the  number,  and  is  always  positive;  and 

(2j  In  the  case  of  numbers  less  titan  unity,  tJte  characteristic 
is  one  more  than  the  nutnber  of  cyphers  after  the  decimal 
point  in  the  number,  or  is  the  same  number  as  the  place 
from  the  decimal  point  which  the  first  significant  figure 
occupies;  and  is  alivays  negative. 

The  usual  published  tables  of  common  logarithms  give  the 
mantissa  for  each  number  from  unity  to  108,000,  and  the  logs, 
of  all  numbers  containing  5  figures  may  be  found  at  one 
reference.  If  the  number  of  which  the  log.  is  required  contains 
more  than  5  figures,  the  corrected  log.  is  found  by  reference 
to  one  of  the  tables  of  proportional  parts  given  in  the  margin 
of  the  tables,  but  all  the  published  tables  describe  so  fully  how 
this  is  done  that  it  is  not  necessary  to  repeat  it  here. 

There  are  also  several  other  practical  operations  required 
which  are  fully  explained  in  the  tables,  amongst  others,  (1) 
finding  the  antilog.  or  the  number  corresponding  to  any 
logarithm,  and  (2)  the  method  of  dealing  with  logs,  having 
negative  characteristics,  either  by  addition  or  subtraction, 
which  follows  the  ordinary  rules  of  algebra. 

Special  attention  should,  however,  be  given  to  the  rules  as 
to  multiplying  or  dividing  a  log.  with  a  negative  characteristic. 
The  following  method  of  dividing  such  a  log.  is  used  by  the 
author  in  order  to  find  the  value  of  the  factor  R,  and  differs 
from  the  method  given  in  the  tables,  but  is  simpler.  It  is  as 
follows  :  — - 


LOGARITHMS 


27 


Having  obtained  the  log.  of  EN  (N  =  20  years),  viz.,       2-987  8003 
it  is  required  to  divide  the  log.   by  20,  in  order 
to  obtain  log  R-, 

proceed  by  adding  20,     20" 


=  18-987  8003 


Divide  this  log.  by  20=   0949  3900 
and    deduct    1,    to    correct    the    addition    of   20, 

divided  by  20,  =    l' 


Leaving  tlie  required  log.       1'949  3900 


It  is  sometimes  required  to  divide  one  log.  by  another,  as  in 
Calculation  XXXII,  E.,  in  order  to  find  the  number  of  years,  N, 
in  an  equated  period  at  a  given  rate  per  cent.,  knowing  tbe 
value  of  the  factors  E^  ^nd  E.  If  both  the  logs,  are  positive  or 
negative,  they  may  be  treated  as  ordinary  numbers  and  tlie 
corresponding  logs,  found  in  the  usual  way,  but  if  their  charac- 
teristics are,  one  plus  and  the  other  minus  in  sign,  they  must  be 
reduced  to  the  same  sign. 


28  REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 

CHAPTER  III. 

SIMPLE  AND  COMPOUND  INTEREST. 

Simple  Interest.  An  arithmetical  progression.  Formula. 
Tables.     Incidental  use  of  the  Tables. 

Compound  Interest.  A  geometrical  progression.  Derivation 
OF  THE  Formula,  A  =  P  R^,  relating  to  Compound 
Interest,  from  the  algebraical  formula,  l  =  ar  ^"~'^\ 
relating  to  a  geometrical  progression.  Explanation 
OF  terms.     Difference  between  the  amounts  of  £1  and 

OF    £1     per    annum    at    the    end    of     1    YEAR.        "  PrESENT 

Value  "  compared  with  "  Practical  Discount." 

Simple  Interest,  an  Arithmetical  Progression.  Simple 
interest  is  an  arithmetical  progression,  and  the  amount  of  any 
sum  of  money,  at  the  end  of  any  given  term,  may  be  ascertained 
by  continued  addition  of  the  interest  upon  the  sum  for  one  year, 
or  other  period,  at  the  stated  rate  per  cent.  It  is  the  method 
in  general  use  in  all  commercial  and  financial  transactions, 
although  in  cases  where  balances  in  an  account  current  are 
struck  at  stated  j^eriods,  it  may  partake  of  the  nature  of 
compound  interest.  The  main  feature  of  this  method  is  that 
the  calculations  may  relate  to  varying  sums,  varying  times,  and 
varying  rates  per  cent.,  and  are  expressed  by  the  formula  : — - 

Principal  X  rate  per  cent,  per  annum  x  years 


Interest, 


100 


and  the  ascertained  amount  of  interest  is  stated  in  the  same 
terms  as  the  principal,  whether  pounds  sterling,  shillings 
sterling,  dollars  or  other  currency.  All  such  calculations  are 
extremely  simple,  and  many  tables  are  published  giving  the 
amounts  of  interest  on  varying  amounts  of  principal  for  varying 
periods,  whether  days  or  years.  The  above  formula  is  the  one 
used  to  calculate  the  amount  of  interest  for  one  or  more  years. 
If  it  be  required  to  calculate  the  amount  of  interest  for  any 
number  of  days  at  a  given  rate  per  cent,  per  annum,  the 
formula  becomes  :  — 

_    ,  ,        Piiiicipwl  X  rate  i)er  colli,  per  aiinuiu  X  iiunibfM' of  diivs 

Interest,  = 1 _ i_ - 

100  X  865 


SIMPLE  AND  COMPOUND  INTEREST  29 

The  utility  of  any  table  of  simple  interest  is  limited  only 
by  its  size,  and  it  is  very  easy  by  means  of  tbe  above  formula 
to  ascertain  any  required  sum  not  given  in  the  table,  Tliere  are 
several  modifications  of  tbis  method  to  suit  individual  or  special 
requirements  which  do  not,  however,  require  special  mention. 

Whilst  on  the  question  of  simple  interest,  there  is  an 
interesting  method  of  using  such  tables  which  may  not  be 
generally  known.  It  is  often  required  to  ascertain  the  amount 
of  rent,  or  other  annual  sum  for  a  given  number  of  days.  If, 
for  instance,  it  is  required  to  ascertain  the  amount  of  97  days' 
rent  at  £865  per  annum,  proceed  as  follows  :  — Multiply  the 
annual  rent  £865,  by  20  =  £17,300;  refer  to  the  tables  and 
ascertain  97  days'  interest  upon  £17,300  at  5  per  cent.  This 
will  be  the  amount  of  97  days'  rent. 

Similarly,  the  annual  rent  may  be  multiplied  by  25  and 
interest  upon  the  product  ascertained  at  4  per  cent.,  but  the 
above  method  is  the  simplest  as  it  involves  multiplying  by  2 
only.  As  a  matter  of  fact  any  other  equivalent  multiplier  and 
rate  per  cent.,  having  100  for  their  product,  may  be  used. 
This  method  may  be  applied  to  ascertain  the  proportion  of  the 
annual  sinking  fund  instalment  to  be  set  aside  in  respect  of  a 
loan  borrowed  at  various  dates  in  one  year  as  afterwards  pointed 
out  in  Chapter  XXX. 

Compound  Interest,  a  Geometrical  Progression.  Com- 
pound interest  differs  from  simple  interest  in  that  it  is  a 
geometrical  progression  in  which  the  rate  per  cent,  is  always 
uniform  during  the  whole  period,  and  the  periods  are  all  equal, 
whether  years,  half  years,  months,  or  otherwise.  There  are 
several  published  tables  of  compound  interest,  and  many  tables 
have  been  calculated  for  special  purposes.  The  one  most 
generally  used  in  England  is  by  William  Inwood  (18th  Edition 
published  1880),  commonly  referred  to  as  "  Inwood's  Tables." 
A  new  and  much  improved  edition  was  issued  in  1899,  revised 
and  extended  by  Mr.  William  Schooling. 

Tables  of  this  character  are  extremely  useful,  and  provide 
for  the  majority  of  calculations  required  to  be  made  by  Local 
Government  and  municipal  authorities,  actuaries,  accountants, 
bankers  and  valuers,  and  the  officials  of  commercial  and  financial 
undertakings. 

Derivation  of  the  Formitlj..  It  is  a  very  interesting 
study  to  analyse  the  tables  mathematically  and  to  derive  each 


30  REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

tahle   from   the   simple  algebraical    formula    used   to   find   the 
last  of  a  series  of  numbers  in  a  geometrical  progression,  viz. :  — 

I  =  ar'^-  ^ 
where  a  =  the  first  term, 
/  =  the  last  term, 
r  =  the  constant  factor  or  ratio, 
w  =  the  number  of  terms  in  the  progression. 

A  geometrical  progression  consists  of  a  series  of  numbers 
which  increase  or  decrease  by  a  constant  factor  or  common 
ratio,  and  many  problems  may  be  solved  by  means  of  the 
algebraical  formulae  relating  to  such  a  progression,  namely,  the 
sum  of  a  series,  either  finite  or  to  infinity,  the  insertion  of  a 
number  of  geometric  means  between  two  numbers,  and  finding 
the  last  term  of  a  series.  Problems  involving  compound 
interest,  however,  include  only  the  first  term,  the  ratio,  and  the 
last  term,  all  of  which  may  be  determined  by  means  of  the 
algebraical  formula  with  only  slight  modification.  The  factors 
(which  remain  unchanged  except  as  regards  the  actual  vSymbol) 
are  as  follows  : —  • 

n  =  fhc  frst  tenii  of  the  i^rofjrc^f^inn,  which  corresponds  to  the 
principal  sum  (P)  at  the  beginning  of  the  number  of 
years. 

l=t]ie  last  terin  of  the  progression,  which  corresponds  to  the 
amount  (A)  of  the  principal  sum  (P)  at  the  end  of  the 
number  of  years. 

r  =  the  common  ratio,  or  the  number  by  which  each  term  in  the 
progression  is  multiplied  in  order  to  find  the  succeeding 
term.  In  the  formulse  relating  to  compound  interest 
this  is  expressed  by  the  symbol  (R)  because  when  dealing 
with  annuities,  a  symbol  is  required  to  represent  a  new 
factor  (R-1)  which  is  denoted  by  (r),  and  which  will  be 
explained  later. 
11  =  file  nuinher  of  terms  in  the  progression,  and  is  the  only 
factor  in  the  algebraical  formula  requiring  any  alterati<m 
in  the  sense  in  M-hich  it  is  used.  In  both  formuht  the 
ratio  acts  in  exactly  the  same  manner,  or  once  during 
each  interval  in  the  progression,  and  it  acts  upon  each 
term  except  the  last.  In  any  progression,  the  number 
of  intervals  between  the  terms  is  one  less  than  the  number 
of  terms,  or,  as  it  may  be  expressed  : 

(??)  intcrvals=(» -1)  terms. 


SIMPLE  AND  COMPOUND  INTEREST  31 

In  the  case  of  compound  interest,  tlie  intervals  are  years, 
or  other  equal  periods  of  time,  consequently  the 
algebraical  formula  is  altered  by  substituting  (N)  years 
for  ('i  — 1)  terms,  using  the  capital  (N)  to  denote  the 
number  of  years  in  order  to  distinguish  it  from  the  small 
(7?)  which  denotes  the  number  of  terms  in  the  algebraical 
formula . 

Substituting  the  amended  symbols  as  above, 

I  =  ar'"' ' ^  becomes,  A  =  P  R^, 

and  the  above  symbols  have  the  following  meaning  throughout 
the  book  :  — 

A  =  the  auwtint,  or  the  ultimate  sum  to  which  the  present  sum 
(P)  will  accumulate  in  (N)  years  at  the  ratio  or  constant 
factor  (R).  This  symbol  will  be  used  to  denote  this 
factor  whether  it  represents  the  ultimate  sum  required  to 
be  found  at  the  end  of  the  period ;  or  the  given  sum  due 
at  the  end  of  the  period,  of  which  it  is  required  to  find 
the  present  value  (P). 

The  use  of  the  word  "  amount "  is  different  from  the  usual 
meaning  attached  to  it  in  ordinary  language,  and  it  is  very 
necessary  to  distinguish  it  from  a  sum  of  money.  A  very  much 
better  word  would  be  "  accumulate." 

'P  =  thc    ptincipal    or    j^resent    value,    and    denotes    a    sum    of 
money  in  hand,  or  due,  now.     It  also  denotes  : 
(1)  the  present  value  of  a  definite  sum  of  money  (A)  due  at 

the  end  of  a  stated  period  of  years,  and 
(2)  the  present  value  of  an  annuity  or  other  periodic  sum 
(A^)  payable  or  receivable  at  the  end  of  each  of  a  stated 
number  of  years  or  periods. 

These  two  factors  (A)  and  (P)  are  intimately  related.  (P)  is 
the  first  term,  and  (A)  the  last  term,  of  a  geometrical  progres- 
sion. (P)  is  the  present  value  of  (A)  due  at  the  end  of  a  stated 
term,  and  (A)  is  the  amount  to  which  (P)  will  accumulate 
dviring  that  period. 

'R  =  thc  ratio  or  common  factor,  and  denotes  the  rate  of  increase 
(expressed  in  terms  of  unity)  in  each  term  of  the  pro- 
gression. It  does  not  denote  the  rate  per  cent,  per 
annum,  although  it  is  derived  directly  from  the  rate  per 
cent.     It  is,  in  all  cases,  £1  increased  by  interest  upon 


33  REPAYMENT   OF   LOCAL   AND    OTHER   LOANvS 

£1  for  one  year  at  the  rate  per  cent,  in  question;  in  the 
case  of  5  per  cent,  it  is  105,  as  will  be  clearly  shown  in 
Calculation  (IV)  1,  and  so  on  for  every  other  rate 
per  cent.  The  ratios  corresponding  to  each  rate  per  cent, 
from  I  to  7  per  cent,  are  given  later  in  Table  No.  Y.  A, 
together  with  the  corresponding  logarithms.  In  calcula- 
tions involving  compound  interest  the  actual  rate  per 
cent,  is  never  used,  but  only  in  its  relation  to  £1  by  way 
either  of  a  ratio  (R)  or  of  the  interest  upon  £1  for  one 
year  (r). 

'N  =  the  number  of  years,  or  other  equal  periods,  and,  as  already 
explained,  must  not  be  confounded  with  [n)  in  the 
algebraical  formula  for  a  geometrical  progression  which 
denotes  the  number  of  terms  in  the  progression.  This 
number  of  terms  includes  the  first  term,  but  in  the  case 
of  compound  interest  the  number  of  years  is  one  less 
than  the  number  of  terms  in  the  progression;  therefore 
(N)  years  =(n-l)  terms.  In  the  case  of  annuities,  a 
modification  of  the  above  formula  will  be  required,  the 
derivation  of  which  from  the  formula  (A  =  P  R^)  ^{i\  ^g 
fully  explained.  This  modified  formula  will  contain 
additional  symbols,  namely — 

Ay ^  the  annuity  or  other  periodic  sum,  to  be  set  aside,  paid  or 
received  at  the  end  of  each  year  or  period. 

^l  =  the  amount  of  the  annuity  or  other  periodic  sum  (A//) 
accumulated  for  a  given  number  of  years  or  periods  (N) 
at  a  given  rate  per  cent.  This  symbol  bears  the  same 
relation  to  a  periodic  sum  (A?/)  as  (A)  bears  to  a  present 
sum  (P). 

r  =  the  interest  upon  £1  for  one  year  or  period  at  the  stated  rate 
per  cent.  It  is  found  from  the  above  ratio  or  common 
factor  by  deducting  unity  therefrom.  The  values  of  this 
factor  for  the  various  rates  per  cent.,  and  the  corre- 
sponding logs.,  are  shown  in  Table  No.  Y.  A,  which  will 
be  given  later. 

The  above  formula,  A  =  P  R^,  with  its  various  modifications, 
may  be  used  to  find  factors  which  are  sufficient  to  solve  all 
questions  of  compound  interest  in  relation  to  sinking  funds  and 
annuities.  The  actual  values  of  each  factor  are  capable  of 
l>eing  tabulated  for  varying  terms  at  varying  rates  per  cent. ; 
and  to  make  them  generally  useful  tlu>  results  are  stated  in  the 


SIMPLE  AND  COMPOUND  INTEREvST  33 

published  tables  in  terms  of  £1  so  that  any  problem  as  to  other 
amounts  may  be  solved  by  multiplying  or  dividing  the  actual 
figure  in  the  problem  by  the  amounts  given  in  the  published 
tables.  In  the  old  edition  of  Inwood  these  tables,  I.  to  V., 
are  given  separately;  but  in  the  new  edition,  Tables  I.  to  lY.  are 
shown  in  four  separate  columns  in  one  table.  Throughout  the 
book  they  will  be  referred  to  as  Tables  I.  to  Y.,  and  anyone 
using  the  new  edition  will  refer  to  the  corresponding  column  in 
the  table  on  pages  50  to  85. 

The  Difference  between  the  Amounts  of  £1  and  of  £1 
PER  Annum  at  the  End  of  One  Year.  It  is  important  to 
remember  that  in  all  calculations  involving  (P)  the  sum  of 
money  which  it  represents  is  due  or  in  hand  at  the  beginning  of 
the  first  year  of  the  period.  In  the  case  of  annuities,  the  annual 
sum  is  assumed  to  be  set  aside,  paid,  or  received  at  the  end  of 
the  first  and  every  subsequent  year  of  the  period.  This  is  very 
important,  sufficiently  so  to  justify  the  following  extracts  from 
the  tables : — 

Table      1.     The  amount  (A)  of  (P)  £1  at  the  end 

of  one  year  at  5  per  cent,   is     £1'05 

Table    11.     The  present  value  (P)  of  (A)  £1  due 

at  the  end  of  one  year  at  5  per  cent,  is     £0'9524 

Table  111.     The  amount  (M)  of  [Ky)  £1  per  annum 

at  the  end  of  one  year  at  5  per  cent,  is     £100 

Table  IV .     The  present  value  (P)  of  (Ai/)  £1  per 

annum  for  one  year  at  5  per  cent,  is     £09524 

Prom  the  above  it  will  be  seen  that  the  amount  of  £1  at  the 
end  of  one  year  (£105)  is  greater  than  the  amount  of  £1  per 
annum  at  the  end  of  one  year  (£1)  because  the  £1  is  in  hand 
and  bears  interest  during  the  first  year,  whereas  the  annuity 
of  £1  per  annum  is  not  due  until  the  end  of  the  year.  But  on 
comparing  the  present  value  of  £1  due  at  the  end  of  one  year, 
and  the  present  value  of  £1  per  annum  due  at  the  end  of  one 
year,  they  are  the  same  (viz.,  £0-9524)  because  they  are  both 
due  at  the  same  time. 

Problems  may  arise  involving  a  variation  from  this  principle 
when  dealing  with  purchases  on  the  deferred  payment  system. 
In  such  cases,  the  annual  instalment  of  principal  and  interest 
combined    is   generally   payable    at   the    end    of   the    first  and 


34  REPAYMENT    OF   LOCAL   AND    OTHER    LOANS 

subsequent  years,  in  the  above  manner,  but  it  sometimes 
liappens  that  the  agreement  provides  that  the  first  payment  shall 
be  made  at  the  beginning  of  the  first  year  which  makes  an 
important  alteration  in  the  method.  Such  problems,  however, 
rarely  arise  in  connection  with  the  sinking  funds  of  local 
authorities  or  of  commercial  or  financial  undertakings,  and  will 
not  be  further  considered. 


Practical  Discount  as  Compaeed  with  Present  Value, 
Discount  of  Bills,  &c.  The  above  extracts  show  that  £100  at 
5  per  cent,  at  the  end  of  one  year  will  amount  to  £105,  and  that 
£105  due  at  the  end  of  one  year  at  5  per  cent,  is  worth  noAv 
£100.  The  difference  between  the  two  amounts  viz.,  £5,  is  the 
mathematical  or  true  discount,  and  is  based  upon  the  present 
value.  In  practical  finance  the  method  adopted  in  discounting 
bills  is  to  deduct  interest  at  the  rate  per  cent,  from  the  amount 
of  the  bill  payable  at  the  end  of  the  period,  and  as  this  amount 
is  always  greater  than  the  present  value,  practical  discount, 
as  it  is  called,  is  always  greater  than  the  mathematical  or  true 
discount.  For  instance,  a  bill  for  £105  due  at  the  end  of  one 
year,  and  discounted  by  the  bank  at  5  per  cent.,  is  worth  now 
£99"75,  ascertained  as  follows:  — 

Amount  of  the  bill      £10500 

Less  the  practical  discount  at  5  per  cent,  for  one  year         £5^25 


or  a  net  value  of      £99'75 

If  the  customer  leaves  this  sum  on  deposit  with  the 
bank,  at  5  per  cent,  he  will  at  the  end  of  the 
year  be  credited  with  5  per  cent,  upon  £99"T5  or         £4-9875 


and  will  then  receive      ...     £104*7375 
as  compared  with  the  amount  of  the  bill       £105" 


a  difference  of    £0-2625 


In  other  words,  he  would  lose  and  the  bank  woubl  gain  £02625 
although  the  bank  liav<>  liad  the  use  of  th(>  money  for  the  whole 
of  the  year. 


SIMPLE  AND  COMPOUND  INTEREST  35 

The  bank  would  gain  the   difference  between  the 

practical  discount  of £5"^5 

and  the  true  or  mathematical  discount  of      £500 


£0-25 


And  in  addition,  interest  upon  this  amount  for  one 

year  at  5  per  cent.,  or       ••■         £001-^5 


£0-2625 


This  proves  that  the  present  values  as  given  in  the  tables  of 
compound  interest  are  not  available  for  discounts  which  are 
merely  arithmetical  calculations,  and  for  which  special  tables 
are  constructed. 


36  REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


CHAPTER  ly. 

COMPOUND  INTEREST  AS  APPLIED  TO  A  SUM  OF 

MONEY. 

TABLE  I.     The  amount  of  £1  in  any  number  of  years. 

The  formula,  A  =  P  Ji^,  and  rules  deduced  therefrom. 
Calculation  by  the  arithmetical  method.  Compilation 
OF  Tables.     Thoman's  method  and  formula. 

Author's  .Standard  Calculation  Form,  No.  1. 


Formulae. 

A.  'To  find  the  Amount  of  £1  in  any  number  of  years,  as 

given  in  the  published  tables  :  — 
Formula,  A  =  RN 

by  logs.  :      Log.  (Amount  of  £l)  =  Log.  E^ 

B.  To  find  the  Amount  of  any  sum  of  money  in  any  number 

of  years  :  — 

Formula,  A  =  P  RN 

by  logs:      Log.    [Amount   of  principal   sum)  =  Log. 
[principal  sum)  +  Log.  R^ 

The  above  formuhe,  and  methods  by  logs,  apply  equally  to 
Thoman's  Formulce  and  Tables^  which  are  fully  described  in 
Chapter  IX. 

General  Rules  deduced  from  the  above  formulae. 

To  find  the  amount  of  any  sum  of  money  in  any  number  of 
years.  Author's  Standard  Calrulaf  ion  Form,  No.  1. 

Rule  1.     If  the  rate  per  cent,   be  not  given   in   Table  /,  or  in 
Thoman's  Tables  :  — 

Proceed  by  the  formula  relating  to  Table  I . 

Calculation  {IV)  3  A. 

Rule  2.     If  the  rate  per  ceiit.  be  given  in  Table  I :  — 

Multiply    the   amount   given    in    the    table,    by    the 
given  sum.     The  product  is  the  amount  required. 

Calculation  {/]')3  B. 


THE   AMOUNT   OF   ONE   POUND  37 

Rule  3.     If  the  rate  per  cent,  be  given  in  Thoman's  Tables  :  — 

To  the  log.  of  the  given  sum,  add  the  log.  of  R^  as 

given  by  Tlioman.     The  sum  of  the  logs,  is  the  log. 

of  the  amount  required.  Calculation  (IV)  3  C. 

To  find  the  rate  per  cent.,  or  number  of  years,  proceed  as 

shown    in    the     standard    forui     for     the    purpose,     given     in 

Chapter  X. 


The  formula,  A  =  P  li^,  will  now  be  applied  to  the  solution 
of  problems  involving  compound  interest  in  relation  to  a  sum 
of  money,  whether  now  in  hand  or  payable  or  receivable  at 
any  future  date.     The  published  tables  are  as  follows  :  — 

Table    I.     The  amount  of  £1  in  any  number  of  years. 

Table  II.  The  present  value  of  £1  due  at  the  end  of  any 
number  of  years. 
Each  table  will  be  considered  in  detail  to  show  the  method 
of  compilation  by  means  of  the  above  formula,  but  in  the 
present  case  the  arithmetical  method  of  calculation  will  first 
be  given  in  full,  in  order  to  point  out  the  relation  between  the 
two   methods. 

The  Arithmetical  Method.  In  the  following  calculation, 
IV  (1),  at  the  end  of  the  first  year,  interest  at  5  per  cent,  is 
added  to  the  principal  sum  in  hand  at  the  beginning  of  the 
year.  At  the  end  of  the  following,  and  each  subsequent  year, 
interest  is  added  to  the  amount  of  principal  and  interest 
combined,  at  the  beginning  of  the  year.  The  amount  of  added 
interest  increases  each  year,  but  if  each  item  of  interest  be 
compared  with  the  sum  upon  which  it  is  based,  it  will  be  seen 
that  in  all  cases  they  bear  the  same  ratio,  namely,  005  to  1. 
On  comparing  the  amount  of  principal  and  interest  at  the  end 
of  any  year,  with  the  similar  amount  at  the  end  of  the  succeed- 
ing year,  it  will  be  observed  that  they  are  always  in  the  ratio 
of  1  to  105. 

In  other  words,  although  an  amount  of  interest  has  been 
added  each  year,  the  amount  of  principal  and  interest  at  the 
end  of  each  year  might  have  been  obtained  by  multiplying 
the  amount  at  the  end  of  the  previous  year  by  1-05,  or  the 
ratio  R.  This  is  therefore  a  geometrical  progression  increasing 
at  a  ratio  of  1-05.  This  calculation  will  be  referred  to  again  in 
Chapter  YI,  when  considering  the  derivation  of  the  formula 
relating  to  an  annual  or  other  periodic  payment,  and  the 
discussion  of  the  matter  in  that  chapter  may  be  referred  to  at 
this  stage  with  advantage. 


38 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Calculation  (IV)  1. 

To  find  the  Amouut  of  a  given  Sum  at  the  end  of  a  given  term. 

Table  I. 

Required  the  Amount  of  £1  at  the  end  of  5  years  at  5  per  cent., 
compound  interest. 

By  Arithmetical  Calculation. 

0000 
(r)        -0500 


Principal  Sum  at  the  beginning  of  the  first  year. . . 
1.  First  year's  Interest  thereon = 


(1x1-05) 

2.  Second  year's  Interest  thereon 

(1-05  X  105)      

3.  Third  year's  Interest  thereon 

(1-1025  X  1-05) 

4.  Fourth  year's  Interest  thereon 

(1-1576x1-05) 

5.  Fifth  year's  Interest  thereon 

(1-2155x1-05) 


=  (R)     1 


0500 
0525 


1025 
0551 


1576 
0579 


2155 
0608 


1-2763 


which  is  the  required  amount  at  the  end  of  the  5th  year;  and 
agrees  with  the  amount  given  in  Table  I.  A  further  amplifica- 
tion of  this  calculation  will  be  made  in  Chapter  YI. 

The  Mathematical  Method.  It  very  rarely  happens  that 
calculations  of  compound  interest  are  required  for  so  short  a 
period  as  5  years ;  generally  they  are  for  very  much  longer  periods. 
Consequently  the  arithmetical  method  as  shown  in  the  above 
Calculation  (lY)  1,  becomes  cumbrous  and  liable  to  error, 
and  it  is  imperative  to  adopt  a  shorter  method,  namely,  the 
algebraical  or  mathematical  one,  based  upon  the  formula, 
A  =  P  RN.  Here  it  is  required  to  find  the  amount  A,  knowing 
that :  — 

P  =  l,  R  =  105  and  N  =  5. 

The  equation  therefore  becomes  : 

A  =  RN  or  A=  (105)5. 

but  to  raise  R,  or  1"05  to  ihv  5th  power  or  perliaps  to  the  20th, 
30th,  or  60th  power  is  a  much  longer  task  than  to  make  the 


THE   AMOUNT   OF   ONE   POUND  39 

original  calculation  by  the  arithmetical  method,  as  in  the 
previous  example,  and  recourse  is  had  to  logarithms,  which 
have  been  fully  described  in  Chapter  II.  The  calculation  will 
be  made  upon  standard  calculation  form  No.  1  by  method  (A) 
therein  contained,  and  it  will  be  found  that  the  resulting 
amount  agrees  with  the  value  given  in  Table  I  in  the  published 
tables.  It  will  also  be  seen  that  the  resulting  log.  of  the 
required  amount  agrees  with  the  log.  of  H^  in  Thoman's  tables. 

The  above  methods  will  now  be  applied  to  the  following 
example  in  order  to  demonstrate  that  the  calculation  by  means 
of  logarithms  and  the  above  formula  is  quite  as  simple,  not 
only  for  any  longer  period,  but  at  any  rate  per  cent.,  whereas 
the  calculation  by  the  arithmetical  method  will  be  longer  in 
proportion  to  the  number  of  years,  and  will  consequently  involve 
a  greater  possibility  of  error  in  the  arithmetical  computation. 

"  Required  the  amount  of  £500  at  the  end  of  20  years  at 
5  per  cent,  per  annum  compound  interest."       Calculation  (IV)  3. 

As  in  the  previous  calculation,  relating  to  £1  only,  the 
result  will  be  ascertained  by  the  same  methods,  viz:  — 

A.  by  the  formula,  A=P  EN     Eule  1 

B.  by  the  published  table  No.  I,  giving  the  amount 

of  £1  at  the  end  of  any  number  of  years Rule  2 

C.  by  Thoman's  tables Rule  3 

in  each  case  adopting  the  logarithmic  method  of  calculation. 
The  above  rules  and  formulse  are  fully  set  out  in  the  heading  to 
this  chapter. 

Thoman's  Method  and  Formula.  Altliough  Thoman's 
method  applies  more  particularly  to  calculations  involving 
annuities  or  other  periodic  payments,  these  tables  may  with 
advantage  be  utilised  to  solve  problems  relating  to  the  amount 
and  present  value  of  £1,  owing  to  the  fact  that  the  actual  logs, 
of  RN  are  there  given,  instead  of  having  to  be  taken  from  the 
loo;,  tables.  The  full  consideration  of  Thoman's  tables  is 
contained  in  Chapter  IX. 


40 


REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


Calculation   (IV)   2. 

Standard  Calculation  Form,  No.  1. 

To  find  the  future  amount  of  a  present  sum,  and  thereby  prove 
the  accuracy  of  the  published  table.  Table  I. 

Required  the  amount  of  £1  at  the  end  of  5  years  at  5  per  cent., 
per  annum,  compound  interest. 


(A)     Bj 

^  Fornmla.                  A  =  P  R^ 

Rulel, 

Chapter  IV. 

Log.. 

'  Log.  Ratio 

multiply  Log.  R  by 

R 

105 
5 

00211893 
5 

R^ 

RN 

(1-05)5 

0-1059465 

Log.  Present  Sum 
add  Log.  RN  above 

P 

RN 

r 

0- 

0-1059465 

A 

0-1059465 

Required  future  amount,  £1-27628,  which  agrees  with  the  result 
obtained  by  the  arithmetical  method,  Calculation  (IV)  1, 
and  also  with  the  amount  given  in  Table  I. 


(B)     By  Table  I.                    A  = 

P  RN 

Rule  2, 

ChapterlV. 

Table  I.     5  years,  5  per  cent. 
Amount  of  £1 

add  Log.  Present  Sum 

RN 

p 

1-27628 

A 

1-27628 

Required  future  amount,  £1 

•27628, 

as  given  in 

Table  I. 

(C)      By  Thoman's  Table.       A  = 
5  per  cent.  5  years. 

P  RN 

Rule  3, 

Chapter  IV. 

Log.  Present  Sum 
add  Log.  RN 

P 

RN 

1- 

0- 

0-1059465 

A 

01059465 

Required    future    amount,    £1-27628.     This    log.    is    given    in 
Thoman's  Table. 


THE   AMOUNT   OF   ONE   POUND 


41 


Calculation    (IV)   3. 

Standard  Calculation  Form,  No.   I. 

To  find  the  future  amount  of  a  present  sum. 


Table  I. 


Required  tlie  amount  of  £500  at  the  end  of  20  years  at  5  per 
cent,  per  annum,  compound  interest. 


(A)     By  Fornuda.                  A  =  : 

[>   l.N 

Rulel, 

Chapter  lY. 

Log-. 

rLo^.  Ratio 

multiply  Log.  R  by 

Log.  Present  Sum 
add  Log.  RN  above 

R 

N 

1-05 
20 

00211893 
20 

R'^ 

RN 

(1-05)20 

0-4237860 

P 

RN 

500 

2-6989700 
0-4237860 

A 

3-1227560 

Required  future  amount,  £1326-65. 


(B)     By  Table  I.                    A  = 

P  RN 

Rule  2, 

Chapter  lY. 

Table  I.     20  years,  5  per  cent. 
Amount  of  £1 

add  Log.  Present  Sum 

RN 

p 

2-6533 
500 

0-4237860 
2-6989700 

A 

3-1227560 

Required  future  amount,  £1326-65. 


(C) 

By  Thoman's  Table .       A  =  P  RN 
5  per  cent,  20  years. 

Rule  3, 

Chapter  lY. 

Log.  Present  Sum 
add  Log.  RN 

P 

RN 

500 

2-6989700 
0-4237860 

A 

31227560 

Required  future  amount,  £1326' 65. 


42     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


CHAPTEE  V. 

COMPOUND  INTEREST  AS  APPLIED  TO  A  SUM  OF 
MONEY  (Continued). 

TABLE  II.     The  present  value  of  i^i,  due  at  the  end  of  any 
number  of  years. 

Derivation  of  the  formula,  P=  ^^    and  rules  deduced 

THEREFROM.  COMPILATION  OF  T AISLES.  TaBLE  OF  EaTIOS. 
AND  LOGS..  OF  E,  AND  r.  CALCULATIONS  FOR  PERIODS  OTHER 
THAN    YEARS.       ThOMAN's    METHOD    AND    FORMULA. 

Author's  Standard  Calculation  Form,  No.  2. 


Formulae. 

A.  To  find  the  present  value  of  £1,   due  at  the  end  of  any 

numher  of  years,  as  given  in  the  puhlished  tables:  — 

Formula,  P=^p^- 

hy  logs.:      Log.  [present  value  of  £T)  — 
Log.  1{  =  0)- Log.  R^ 

B.  To  find  the  present  value  of  any  s^im  of   money,  due  at 

the  end  of  any  numher  of  years  :  — 

Form  ula,  P  =  pj^ 

by  logs.:      Log.  (present  value)  =  Log.  [amount  d'ue 
at  end  of  period) -Log.  E^ 

The  above  formulce,  and  methods  by  logs.,  apply  equally  to 
Thoman's  formula'  and  tables,  which  are  fully  described  in 
Chapter  IX. 


General  Rules  deduced  from  the  above  formulae. 

To  find  the  present  value  of  any  sum  of  money,  due  at  the 
end  of  any  number  of  years. 

Author's  Standard  Calculation  Form,  No.  2. 


THE  PRESENT  VALUE  OF  ONE  POUND       43 

Rule  1.     If  the  rate  per  cent,  be  not  given  in  Table  II,  or  in 
Thoman's  Tables:  — 

Proceed  by  the  formula  relating  to  Table  II. 

Calculation  (F)  2  A. 

Rule  2.     If  the  rate  per  cent,  be  given  in  Table  II:  — 

Multiply  the  amount  given  in  the  table,  by  the 
given  sum.  The  product  is  the  present  value 
required.  Calculation  (Y)  2  B. 

Rule  3.     If  the  rate  per  cent,  be  given  in  Thoman's  Tables  :  — 

From  the  log.  of  the  given  sum,  deduct  the  log.  of 

RN  as  given  by  Thoman.     The  remainder  is  the  log. 

of  the  present  value  reqtiired.        Calculation  (\')  2C. 

To  find  the  rate  per  cent,  or  number  of  years,  proceed  as 

shown  in  the  standard  form  for  the  purpose,  given  in  Chapter  X. 


Derivation  of  the  Formula.  Having  ascertained  the 
methods  of  finding  the  accumulated  amount  of  any  sum  of 
money  at  the  end  of  any  number  of  years  at  any  rate  per  cent., 
the  converse  will  now  be  considered,  namely,  the  present  value 
of  any  sum  due  at  the  end  of  a  given  number  of  years.  These 
two  factors  are  intimately  related.  In  Calculation  (IV)  3  it 
was  found  that  a  present  sum  of  £500  at  5  per  cent,  compound 
interest  will  in  20  years  amount  to  £1326-65,  but  this  denotes 
also  that  £500  at  5  per  cent,  is  the  present  value  of  £1326-65 
payable  at  the  end  of  20  years,  consequently  the  formula 
A  =  P  E^  will  give  two  results,  or  reciprocals,  namely, 
Table    I.     The   amount   of   a   given   sum,    P,    in    any 

number  of  years,  N, =A. 

Table  II.     The  present  value  of  a  given  sum,  A,  due 

at  the  end   of   any  number  of  years,    N,      =P. 

The  formula  for  finding  the  present  value  of  a  given  sum, 
instead  of  being, 

A  =  P  RN,  becomes  P=  ^^^ 

or,  in  other  words,  the  present  value  of  a  sum  due  at  a  future 
date  may  be  ascertained  by  dividing  the  amount  due  at  the 
end  of  the  number  of  years  by  the  ratio,  R,  raised  to  the  power 
equal  to  the  number  of  years,   N.     In  the  case  of  £1   as  in 

Table  II,  the  formula  becomes  P=  .^  since  A,  the  future 
amount,   is  £1. 


44  REPAYMENT    OF    LOCAL    AND    OTHER   LOANvS 

This  formula,   P=  ^pj^    may  now  be  used  to   ascertain  the 

amounts  given  in  Table  II,  and,  as  in  the  previous  example, 
the  calculation  will  be  made  by  three  different  methods,  namely, 

A,  by  formula Rule  1. 

B,  by    the     published     Table     II,   giving     the 

present  value  of  £1  due  at  the  end  of  any 

number  of  years     Rule  2 . 

C,  by  Thoman's  Tables Eule  3. 

in  each  case  adopting  the  logarithmic  method  of  calculation. 
The  above  rules  and  formulae  are  fully  set  out  in  the  heading 
of  this  chapter. 

Thoman's  Method  and  Formula.  In  considering  the 
methods  of  finding  the  amounts  of  £1  in  any  number  of  years 
as  given  in  Table  I,  attention  was  drawn  to  the  advantage  of 
using  Thoman's  tables.  It  was  found  that  the  calculation 
by  this  method  is  similar  to  the  calculation  by  Table  I,  but  in 
the  case  of  Table  II,  relating  to  the  present  value  of  a  future 

sum,  it  is  necessary  to  make  use  of  the  reciprocal  of  R^,  or  p^. 

The  only  difference  between  the  two  tables  is  that  in  the  case  of 
Table  I  the  log.  of  R^  is  added  to  the  log.  of  the  present  sum, 
whereas  in  the  case  of  Table  II  the  same  log.  is  deducted  from 
the  log.  of  the  future  given  sum  of  which  it  is  required  to  hnd 
the  present  value.     Calculation  (V)  1. 

The  same  formula  will  next  be  used  in  order  to  ascertain 
the  present  value  of  £1326"  65  due  at  the  end  of  20  years  at 
5  per  cent,  compound  interest,  and  thereby  prove  the  converse 
of  Calculation  (lY)  3,  adopting  the  same  methods,  namely, 
by  formula :  by  the  published  Table  II :  and  by  Thoman's 
method.     Calculation  (Y)  2. 

Calculations  for  Periods  other  than  Years.  In  cases 
where  it  is  required  to  calculate  compound  interest  for  periods 
oth(>r  than  years,  and  the  rate  per  cent,  is  expressed  as  per 
annum,  it  is  necessary  to  take  a  rate  per  cent,  proportionate  to 
the  period  of  a  year.  For  instance,  if  it  be  required  to  calculate 
the  amount  of  a  sum  of  money  rolling  \i\)  half  yearly,  douhle 
the  nu  lit  her  of  years  anil  take  one-half  the  rate  per  cent,  per 
annum,  as  follows:  — 


THE  PREvSENT  VALUE  OF  ONE  POUND       45 

£1  at  the  end  of  10  years  at  10  per  cent,  per  annum 

will  amount  to  (yearly  breaks) £2'5937 

£1  at  the  end  of  10  years  at  10  per  cent,  per  annum 

will  amount  to  (half-yearly  breaks) 

=  20  years  at  5  per  cent £2"6533 

£1  at  the  end  of  10  years  at  10  per  cent,  per  annum 

will  amount  to  (quarterly  breaks) 

=  40  years  at  2i  per  cent £2"6851 

This  is  a  very  useful  method  to  adopt  when  it  is  required 
to  ascertain  the  effect  of  compounding  the  interest  at  various 
periods,  and  the  rule  applies  equally  to  calculations  involving 
annuities.  x\ll  that  is  necessary  is  to  deal  with  the  number  of 
periods  at  the  corresponding  rate  per  cent,  per  period,  based 
upon  the  rate  per  cent,  per  annum. 

The  Factors  E  (Eatio)  and  r  (The  Interest  of  £1  for  One  Year) 
AND  THE  Corresponding  Logarithms. 
In  order  to  simplify  the  method  by  formulae  and  logs,  the 
following  Table,  Xo.  Y.  A.,  has  been  prepared.     It  gives  the 
ratio  (E)  and  the  corresponding  logs,  for  49  rates  from  i  per 
cent,  to  T  per  cent.     It  also  contains  the  A-alues  and  correspond- 
ing logs,  of  the  factor  {r)  which  is  the  interest  upon  £1  for  one 
year.     There  is  not  anything  difficult  in  the  compilation  of  the 
table,  which  is  here  given  only  for  convenience  of  reference. 
The  logarithms  corresponding  to  any  rate   per   cent,  may  be 
ascertained   from  the  log.   tables   at   the  time   of  making  the 
calculation,  but  since  many  of  the  ratios  contain  six  figures,  it 
involves  the  use  of  the  proportional  parts  of  the  logarithms,  and 
a  reference  to  this  table  will  save  time.     When  dealing  with 
annuities,    the    logs,    of    (E^)    and    {r)    are    required  in  each 
calculation,    and    as    they    have   always    to    be    looked    for    in 
different  parts  of  the  log.  tables,  it  is  a  convenience  to  have 
them  in  one  place.     If  it  is  necessary  to  make  a  calculation  at 
any  intermediate  rate  per  cent,  not  included  in  this  table  all 
that   is  required    is    to    find   the    ratio,    which    is    one    pound, 
increased  hij  interest  vpon  one  jjound,  for  one  year,  at  the  given 
rate  per  cent.,  and  then  the  corresponding  log.     The  factor  (/•), 
as  will  be  seen  from  the  table,  is  ascertained  by  deducting  1 
from  the  ratio  so  found.     The  logs,  of  both  are  found  from  the 
tables  of  logs,  in  the  usual  way,  paying  due  attention  to  the 
sign  of  the  "  characteristic"  of  the  log.  of  (r).     The  logs,  of 
(EN)  are  given  in  Thoman's  tables  for  many  rates  per  cent,  for 
a  large  number  of  years. 


46 


REPxWMENT  OF  LOCAL  AND  OTHER  LOANS 


Calculation  (V)  1. 

Standard  Calculation  Forni^  ,\o.  2. 

To  find  the  present  value  of  a  sum  due  at  the  end  of  any  number 
of  years,  and  thereby  prove  the  accuracy  of  the  published 
table.  Table  II. 

Required  the  present  value  of  £1,  due  at  the  end  of  20  years, 
at  5  per  cent,  per  annum  compound  interest. 


(A)     By  Formula.                       "^"IP 

Rule  1, 

Chapter  V. 

Log.  Ratio 
Log-,       viultiply  Log.  R  by 
R^ 

i; 

RN 

105 
20 

(1-05)20 

00211893 
20 

0-4237860 

Log.  Future  Sum 
deduct  Log.  R^  above 

A 

RN 

1- 

0- 

0-4237860 

P 

1-5762140 

Required    present    value,    £037689,    Avhich    agrees    with    the 
amount   given   in   Table   II. 


(B)     By  Table  II. 


R^ 


Rule  2,  Chapter  Y. 


Table  II.     20  years,  5  per  cent. 
Present  value  of  £1 

add  Log.  Future  Sum 


1 

RN 
A 


0-37689 


0-37689 


Required  present  value,  £0-37689,   as  given  in  Table  II. 


(C)     By  Thoman's  Table. 


W 


Rule  3,  Chapter  V 


5  per  cent.  20  years. 

Log.  Future  Sum 
deduct  Log. 

A 

RN 

1- 

0- 

0-4237860 

P 

1-5762140 

Required  present  value,  £0-37689.     This  log.  is  given  in 
Thoman's  Table. 


THE  PRESENT  VALUE  OF  ONE  POUND 


47 


Calculation  (V)  2. 

Standard  Calculation  Form,  No.  2. 

To  find  the  present  value  of  a  sum  due  at  the  end  of  any  number 
of  years.  Table  II. 

Required   the  present  value   of  £132665,   due   at   the  end   of 
20  years,  at  5  per  cent,  per  annum  compound  interest. 


(A)     By  Formula.                       ^^W 

Rule  1, 

Chapter  Y . 

Log.  Ratio 
Log.        multiply  Log.  R  by 

V 

RN 

105 
20 

(1-05)20 

0  0211893 
20 

0-4237860 

Log.  Future  Sum 
deduct  Log.  R^  above 

A 

RN 

1326-65 

3-1227560 
0-4237860 

Required  preseni 

P 

value 

,  £50000. 

2-6989700 

(B)     By  Table  II.                       ^=W 

Rule  2, 

Chapter  Y. 

Table  II.     20  years,  5  per  cent. 
Present  value  of  £1 

add  Log.  Future  Sum 

1 

R^ 
A 

0-37689 
1326-65 

1-5762140 
3-1227560 

P 

2-6989700 

Required  presen 

;  value 

,  £50000. 

(C)     By  Thoman's  Table.          P=  -^ 
5  per  cent.  20  years. 

Rule  3, 

Chapter  Y . 

Log.  Future  Sum 
deduct  Log. 

A 

RN 

1326-65 

3-1227560 
0-4237860 

P 

2-6989700 

Required  present  value,  £50000. 


48 


REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


TABLE  V,  A. 
Giving  the  values  of  (R)  and  (r)  for  the  following  rates  per 
cent,  (from  5^  to  7  per  cent.)  and  the  corresponding  log 
of  each  value. 
(R)  =  the  amount  of  £1  plus  one  year's  interest  at  any  rate 
per  cent. 

=  {l  +  r) 

(r)  =  the  interest  upon  £1  for  one  year  at  any  rate  per  cent. 

=  (Il-l). 

The  Logarithms  of  (R^)  are  given  in  Thoman's  Tables  under 
each  rate  per  cent. 


Rate 

% 


R 

10025 

1005 

10075 


Ratio  =  R 

Loo-.  R 

000108438 

000216606 
000324505 


101  000432137 

1-015  000646604 

1-01625  000700056 

1-0175  000753442 

1-01875  000806762 


1 

8 

1 
4 

3 

8 

1 
2 

5 

8 

# 

-02 

•02125 

•0225 

•02375 

-025 

•02625 

•0275 

•02875 

•03 

■03125 
•0325 
•0:{375 

■o;{5 
•03625 


000860017 
000913207 
000966332 
001019391 
001072387 
001125317 
0-01178183 
001230985 


■01283722 
01:536396 
■Ol;589006 
■01441552 
•014940:55 
■01546454 


Rate  Interest  on  £1  for  1  year  =  j' 

%  r                          Log.  r 

i  00025           33979400 

i  0005            :!•  6989700 

I  00075           3-8750613 


•01 
•015 
■01625 
•0175 

•01875 


002 

002125 

00225 

002375 

0025 

002625 

00275 

0^02875 

003 

003125 

00:325 

0^  0:3375 

00:55 
003625 


2^0000000 
2^1760913 
2^21085:54 
22430380 
22730013 

23010;500 
2^3273589 
23521825 
2^3756636 
23979400 
24191293 
2-4393327 
2^4586378 

2-4771213 

2^4948500 
2^51 188:34 
2^5282738 
2^5440680 
2^5593080 


THE    PRESENT    VALUE    OF    ONE    POUND 


49 


Rate 


Ratio  =  R  Rate   Interest  on  £1  for  1  year  =  r 

R  Log.  R  %  r  Log.  r 


10375 
1-0.3875 

104 

104125 

10425 

1-04375 

1-045 

104625 

1-0475 

1-04875 

1-05 

1-05125 

1-0525 

1-05375 

1055 

105625 

1-0575 

1-05875 

106 

1-06125 

1-0625 

1-06375 

1-065 

1-06625 

1-06750 

1-06875 

1-07 


001598811 
001651104 

0-01703334 
0-01755501 
0-01807606 
0-01859649 
001911629 
0-01963547 
002015403 
002067197 

0-02118930 
002170601 
002222210 
002273759 
0-02325246 
0-02376672 
0-02428038 
002479342 

0-02530587 
0-02581770 
0-02632894 
002683957 
0-02734961 
0-02785904 
002836788 
0-02887613 
0-02938378 


3 

4 

0-0375 

2-5740313 

7 

8 

0-03875 

2-5882717 

4 

0-04 

2-6020600 

1 

8 

004125 

2-6154240 

1 
4 

0-0425 

2-6283889 

3 

S 

0-04375 

2-6409781 

1 
2 

0-045 

2-6532125 

5 

8 

0-04625 

2-6651117 

3 

4 

0-0475 

2-6766936 

7 
8 

0-04875 

2-6879746 

5 

0-05 

2-6989700 

1 

8 

0-05125 

2-7096939 

1 

4 

0-0525 

2-7201593 

3 

8 

0-05375 

2-7303785 

1 
2 

0-055 

2-7403627 

5 

8 

0-05625 

2-7501225 

3 

4 

00575 

2-7596678 

7 
8 

0-05875 

2-7690079 

6 

0-06 

2-7781513 

1 

8 

0-06125 

2-7871061 

1 

4 

00625 

2-7958800 

3 

S 

006375 

2-8044802 

1 
2 

0065 

2-8129134 

5 

8 

006625 

2-8211859 

3 

4 

0-06750 

2-8293038 

7 
8 

0-06875 

2-8372727 

7 

007 

2-8450980 

50  REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


CHAPTER  VI. 

CUMPOUXD  INTEREST  AS  APPLIED  TO  AN  ANNUAL 
OR   OTHER   PERIODIC   PAYMENT. 

TABLE  in.     The  amount  of  ;^i  per  annum  in  any  number 
of  years. 

Geneeal  remarks  as  to  annuities.    The  relation  between 

THE     AMOUNTS      OF      £1,     AND      OF      £1      PER     ANNUM.  ThE 

arithmetical   METHOD   FURTHER    CONSIDERED.       DERIVATION 

(J^N 2\ 
1  AND     RULES     DEDUCED 

THEREFROM.       COMPILATION   OF    TaBLES.       ThOMAn's   METHOD 

and  formula. 

Author's  Standard  Calculation  Form,  No.  3. 


Formulae. 

A.  To  find  the  amount,  of  £1  ycr  oniinm  in  any  nnmher  of 

years,  as  given  in  the  inihlished  tables  :  — 

/RN ix 

(7)  Formula,  M=  (J^^) 

by  logs.  :      Log.  {amount  of  £1  'per  aniium)  = 
Log.  (RN-l)-Lo^.r 
(2)  By  Thoma7i\s  method: — - 

RN 

Formula,  M= — - 

a" 

by  logs.  :      Log.  (amount  of  £1  per  amium)  = 
Log.  RN+lO-Lo^r.  a« 

B.  To  find   the   amount   of  any   annuity   in   any  number   of 

years  :  — 


(7)  Formula,  M  = 


^H"^) 


by  logs.:      Log.  (amount  of  annuity)  =  Log.  annuity  + 
Log.  {l{^-\)-Log.  T 
{2)   By  Tlionian's  method:  — 

RN 

Form  ula,  M  =  A-?/ 

by  logs.  :     Log.  (amount  of  annuity)  =  Log.  annuity  + 
Log.  W^+10-Log.  a« 


THE  AMOUNT   OF   ONE   POUND   PER   ANNUM  51 

The  present  chapter  deals  only  uiitli  the  formula 
M  =  Ay  (  — ^ —  )'  Th Oman's  method  and  formulec,  are  fully 
described  in  Chapter  IX. 

General  Rules  deduced  from  the  above  formulae. 

To  find  the  ainount  of  any  annuity  in  any  number  of  years. 
Author's  Standard  Calculation  Form,  No.  3. 

Rule  1.     If  the  rate  per  cent,  be  not  given  in  Table  III,  or  in 

TJioman's  Tables:  — 

Proceed  by  the  formula  relating  to  Table  III. 

Calculation  (VI)  2  A. 
Rule  2.     If  the  rate  per  cent,  he  given  in  Table  III : — ■ 

Multiply    the    amount   given  in    the    table,    by    the 
given  annuity.     The  product  is  the  amount  required. 

Calculation{VI)2B. 

Rule  3.  If  the  rate  per  cent,  be  given  in  Thoman's  Tables  :  — 
To  the  log.  of  the  given  annuity,  add  the  log.  of  R^ 
as  given  by  ThoTnan.  Add  10  to  the  sum  of  the  two 
logs.,  and  deduct  therefrom  the  log.  of  a^  as  given 
by  Thoman.  The  remainder  is  the  log.  of  the 
required  amount.  Calculation  {^I)  2C. 

To  find  the  rate  per  cent.,  or  mirnber  of  years ^  proceed  as 
shown  in  the  standard  form  for  the  purpose^  given  in  Chapter  X. 


Annuities  or  other  Periodic  Payments.  All  problems 
relating  to  annual  sums  involve  calculations  of  a  more  complex 
character  than  the  steady  accumulation  of  a  given  sum  of 
money.  Matters  are  complicated  by  the  intrusion  of  a  factor 
representing  an  equal  annual  or  other  periodic  sum,  to  be  set 
aside,  received  or  paid,  at  the  end  of  each  year,  and 
accumulated  at  a  given  rate  per  cent.,  for  a  given  number  of 
periods.  Such  an  equal  annual  or  other  periodic  sum  is  called 
an  annuity,  but  in  this  connection  it  should  be  borne  in  mind 
that  actuarially  the  term  annuity  includes  any  definite  sum  of 
money  to  be  paid  or  received  at  the  end  of  any  given  number 
of  regular  intervals.  There  is  room  for  a  better  word,  but  it 
does  not  matter  so  long  as  it  is  known  what  the  term  includes. 
In  the  following  pages  the  word  annuity  will  be  used  to  denote 
any  equal  sum  payable  at  the  end  of  regular  periods,  except 
that  in  the  case  of  sinking  funds,  the  word  "  instalment  "  or 
"  annual  increment  "  will  be  substituted. 


52  REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 

As  in  the  case  of  a  principal  sum,  an  annuity  or  otlier 
periodic  payment  may  be  expressed  in  terms  of  its  "  amount  " 
or  "  present  value  "  whicli  are  given  in  Tables  III  and  IV 
respectively. 

The  Factors  E  (Eatio)  aub  r  (the  Interest  of  £1  for  One  Year). 
In  all  calculations  involving  a  geometrical  progression  tbe 
predominant  factor  is  the  ratio  which,  in  the  algebraical 
formula,  is  expressed  by  the  symbol,  r.  A  pure  geometrical 
progression  relates  only  to  a  series  of  numbers,  increasing  in  a 
definite  ratio,  similar  to  the  annual  accumulation,  by  way  of 
compound  interest,  of  a  given  sum  of  money  as  described  in 
Chapter  IV,  dealing  with  Table  I.  The  algebraical  formula 
for  a  pure  geometrical  progression  does  not  provide  for  any 
further  addition  to  each  term  of  the  progression.  In  the  case 
of  compound  interest,  however,  the  problem  may  be  complicated 
by  the  annual  or  other  periodic  addition  of  a  definite  sum, 
namely,  the  annuity,  and  it  is  necessary  therefore  to  amend  the 
formula,  A=P  B,^ ,  by  dividing  the  factor,  E,  or  ratio,  into  two 
parts,  namely,  the  actual  algebraical  ratio  and  the  equal  annual 
addition  to  each  term  of  the  progression,  representing  the 
constant  sum  or  annuity  to  be  added  to  each  term.  In  the 
algebraical  formula  the  ratio  is  expressed  by  the  symbol,  r. 
In  the  formula  relating  to  compound  interest  two  symbols  are 
used,   namely  :  — 

E  =  the  common  ratio  existing  between  the  successive  terms 
of  the  progression  irrespective  of  any  periodic  equal 
additions  to  the  progression.  This  factor,  E,  in  the 
formula?  relating  to  compound  interest  is  the  equivalent 
of  the  algebraical  factor  (r). 

7'  =  the  annual  or  other  periodic  sum  added  to  each  term  of 
the  progression,  and  which,  as  regards  the  formula 
relating  to  unity,  represents  the  annual  interest  of  £1 
for  one  year. 

In  this  manner  the  accumulation  of  an  annual  sum  by  way  of 
compound  interest,  cannot  properly  be  considered  a  pure 
geometrical  progression.  It  is  rather  the  sum  of  several 
arithmetical  progressions  in  echelon,  which  accounts  for  the 
difficulty  in  determining  the  rate  per  cent,  by  means  of  the 
formula,  as  will  be  seen  on  reference  to  the  standard  form  for 
the  purpose  given  in  Chapter  X. 


THE  AMOUNT  OF  ONE  POUND  PER  ANNUM     53 

The  Eelation  between  the  Amount  of  £1  and  of  £1  feu 
Annum.  It  is  necessary  to  derive  a  formula  relating  to  the 
amounts  and  the  present  values  of  £1  per  annum  as  given  in 
the  published  tables,  which  formula,  although  based  thereon, 
is  of  a  somewhat  more  complicated  character  than  the  simple 
formula  relating  to  Tables  I  and  II.  The  additional  symbols 
which  will  be  required  have  already,  in  anticipation,  been 
explained  in  Chapter  III. 

Before  proceeding  to  find  such  a  formula  the  subject  will  be 
considered  from  the  point  of  view  of  the  accumulation  of  a 
single  sum  now  in  hand,  as  illustrated  by  Calculation  (IV)  1 
in  Chapter  lY.  It  is  possible  to  ascertain  the  sum  to  which 
an  annuity  will  amount  at  the  end  of  a  stated  period,  by 
treating  each  of  the  annual  payments  separately,  and  finding 
the  sums  to  which  they  will  respectively  amount  at  the  end 
of  the  period,  by  the  method  already  considered  in  relation  to 
Table  I.  The  total  of  these  separate  results  will  represent  the 
sum  to  which  the  whole  annuity  will  amount  at  the  end  of  the 
period  (see  columns  1  to  4  in  the  following  table). 

The  method  is  a  cumbrous  one,  and  therefore  not  practical, 
but  the  working  of  such  a  calculation  is  given  in  order  to 
demonstrate  the  relation  between  Table  I,  giving  the  amounts 
of  £1,  and  Table  III,  giving  the  amounts  of  £1  per  annum. 
It  will  also  emjDhasise  Avliat  has  been  already  pointed  out  in 
Chapter  III,  namely,  the  difference  betAveen  the  amounts  of  £1 
and  of  £1  per  annum  at  the  end  of  any  equal  number  of  years, 
as  also  shown  in  columns  5  and  6  in  the  following-  table. 
This  difference  is  due  to  the  fact  that  in  all  calculations  of  this 
nature  the  sum  of  money  of  which  it  is  required  to  find  the 
amount  at  the  end  of  a  term  of  years,  as  in  Table  I,  is  assumed 
to  be  in  hand  and  to  commence  to  accumulate  at  once,  whereas, 
in  the  case  of  an  annuity,  the  annual  or  other  periodic  payments 
are  assumed  to  be  made  at  the  end  of  each  year  or  period,  at 
which  date  they  begin  to  accumulate.  An  annuity  of  £1  for  a 
given  number  of  years  may,  therefore,  be  considered  as  a  series 
of  sums  of  money,  each  of  which  is  deferred,  both  as  to  the 
date  of  payment  and  of  accumulation,  for  1,  2,  3,  4,  etc.,  years. 
Taking  as  an  example  an  annuity  of  £1  for  10  years  to  accumu- 
late at  5  per  cent,  per  annum,  the  sum  to  which  each  separate 
payment  will  amount  at  the  end  of  the  10  years  will  be  ascer- 
tained by  the  method  adopted  in  Calculation  (IV)  1,  and  after 
deriving  the  formula  relating  to  the  amounts  of  an  annuity,  as 
given  in  Table  III,  the  same  example  will  be  worked  out,  by 
means  of  the  formula,  in  Calculation  (VI)  1. 


54 


REPAYMENT   OF    LOCAL   AND    OTHER    LOANS 


TABLE  VI,  A. 

Showing  the  method  of  fiiiding  the  amount  of  an  annuity  from 
the  figures  given  in' Table  I,  relating  to  a  principal  sum 
and  illustrating  the  relation  between  the  amounts  of  £1 
and  of  £1  per  annum  at  the  end  of  one  year, 

Rate  of  accumulation,  5  per  cent. 


TABLE  I. 

TABLE  IIL 

TABLE 

I. 

Amount 

set       Will 
aside    accu- 

at      Ululate 
end  of      for 
year,    years. 

Amount 
of  each 
annual 
sum  at 
end  of 
10th  year. 

Total  at 

end  of 

each  year 

of  the 
amounts 
in  Col.  3. 

Amount  of 

£1  per  annum 

at  end  of 

each  year 

from 

1  to  10  years. 

Amount  of  £1 

at  end  of 

each  year 

from 

1  to  10  years. 

Total  at 
end  of 

each  year 
of  the 

amounts 

in  Col,  6. 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

10 

0 

1-0000 

1-0000 

1 

1-0000 

— 

— • 

9 

1 

10500 

2-0500 

9 

2-0500 

1 

1-0500 

10500 

8 

2 

11025 

3-1525 

3 

3-1525 

2 

1-1025 

21525 

7 

3 

11576 

4-3101 

4 

4-3101 

3 

1-1576 

3-3101 

6 

4 

1-2155 

5-5256 

5 

5-5256 

4 

1-2155 

4-5256 

5 

5 

1-2763 

6-8019 

6 

6-8019 

5 

1-2763 

5-8019 

4 

6 

1-3401 

81420 

i 

8-1420 

6 

1-3401 

7-1420 

3 

T 

1-4071 

9-5491 

8 

9-5491 

7 

1-4071 

8-5491 

2 

8 

1-4775 

11-0266 

9 

110266 

8 

1-4775 

100266 

1 

9 

1-5513 

12-5779 

10 

12-5779 

9 

1-5513 

11-5779 

12-5779 

11-5779 

In  the  above  table  :  — 

Column  1,  contains  the  year  at  the  end  of  which  each  annual 
sum  is  set  aside,  and  Column  2  the  number  of  years 
for  which  it  afterwards  accumulates. 

Column  3,  is  taken  item  by  item  from  Table  I  (with  the 
exception  of  the  first  item  of  £1)  and  shows  the 
amount  of  each  separate  annual  sum  (beginning 
with  the  10th)  at  the  end  of  the  10th  year  as  if  it 
were  accumulated  separately.  The  total  of  Column  3 
is  the  accumulated  amount  of  the  whole  of  the 
annual  sums  obtained  in  this  manner,  and  agrees 
with  the  amount  given  in  Table  III  and  found  by 
Calculation  (YI)  1. 

Column  4,  gives,  at,  the  end  of  each  successive  year,  the  total  of 
the  previous  items  in  Column  3,  which  is  the  amount 
of  all  the  annual  sums  set  aside  up  to  the  end  of  that 


THE   AMOUNT   OF   ONE   POUND   PER   ANNUM  55 

year.  The  items  in  this  coliiinn  correspond,  year  by 
year,  Avith  the  amounts  of  an  annuity  of  £1  given  in 
Column  5,  which  is  copied  item  for  item,  from 
Table  III,  which  gives  the  amounts  of  an  annuity 
of  £1  for  any  number  of  years. 

Column  6,  contains  the  amounts  of  £1  at  the  end  of  each  year 
from   1   to    10  years,    copied,   item   by   item,    from 
Table  I.     These  figures  correspond  with  eacli  item, 
except  the  first,  in  Column  3. 
Column  7,   contains  the  total  at  the  end  of  each  successive  year 
of  the  previous  items  in  Column  6,  and  might  have 
been  obtained  by  adding  together  the  figures  given 
in  Table  I.     On  comparing  the  totals  of  Columns  3 
and  6,  it  will  be  seen  that  the  total  of  Column  G, 
at  the  end  of  the  10th  year,  is  less  by  £1  than  the 
total  of  Column  3.     Similarly,  if  at  the  end  of  any 
year  the  totals  in  Column  5  are  compared  with  the 
totals    in    Column    7,    the    same    difference   will    be 
found. 
Consequently,   if  it  be  required  to  ascertain  the  accumulated 
amount  of  an  annuity  of  £1  for  10,   or  any  other  number  of 
years,  at  5  per  cent,  per  annum  from  Table  I,  which  gives  the 
amounts    of    £1,    it    may   be    found    by    adding    together    the 
successive  amounts  given  in  Table  I  for  9,  or  one  less  than  the 
specified  number  of  years,  and  increasing  the  sum  so  obtained 
by  £1.     The  sum  of  the  9  amounts  is  the  amount  of  nine  years' 
accumulation  of  £1  per  annum,  and  the  £1  so  added  is  the 
last  annual  sum,  which  does  not  accumulate  at  all  owing  to  its 
being  set  aside  on  the  last  day  of  the  last  year  of  the  term. 

Derivation  of  the  Formula.  The  above-described  arith- 
metical method  of  finding  the  amount  of  an  annuity  for  any 
number  of  years  depends  upon  treating  each  annual  sum  as  a 
separate  entity,  but  does  not  treat  the  annuity  qua  annuity, 
and,  further,  it  does  not  give  any  clue  to  a  rule  or  formula 
by  which  the  result  may  be  obtained  by  direct  mathematical 
calculation.  Many  problems  contain  factors  involving  the 
accumulation  of  £1,  and  also  of  £1  per  annum,  and  it  is 
advisable  therefore  that  all  formulae  should  be  expressed  in  the 
same  or  similar  terms.  The  formula  relating  to  the  aiuount 
of  £1  in  any  number  of  years,  namely,  A  =  P  E^,  has  already 
been  ascertained,  and  it  will  now  be  used  in  order  to  deduce 
therefrom  a  formula  relating  to  the  accumulation  of  periodic 
sums.     The  practical  application  of  that  formula  will  be  first 


=,6 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


considered,  and  will  be  based  upon  the  arithmetical  Calculation 
{L\)  1,  as  afterwards  proved  by  means  of  the  formula,  in 
Calculation  (lY)  2.  As  explained  in  Chapter  lY,  at  the  end 
of  the  first  year,  interest  at  5  per  cent,  per  annum  was  added 
to  the  original  principal  sum  of  £1,  and  at  the  end  of  each 
subsequent  year  interest  at  5  per  cent,  per  annum  was  added  to 
the  amount  of  principal  and  interest  at  the  beginning  of  such 
year.  In  Calculation  (lY)  1,  the  interest  added  each  year  was 
treated  as  one  sum,  and  was  not  divided  in  order  to  differentiate 
between  the  interest  added  yearly  in  respect  of  the  original 
principal  sum  as  distinguished  from  the  interest  added  yearly 
upon  the  interest  added  in  previous  years.  In  the  following 
table  (No.  YI,  B)  such  a  distinction  has  been  made,  and  the 
results  obtained  in  Calculation  (lY)  1  are  repeated  in  Column  2. 
The  interest  added  each  year  has  been  divided  as  between  the 
principal  and  the  interest  previously  added,  and  Column  3 
contains  the  constant  annual  amount  of  interest  upon  the 
original  principal  of  £1,  which  is  (/■)  in  the  list  of  symbols 
given  in  Chapter  III.  Columns  4,  5,  6,  and  7  contain  each 
year's  accumulated  interest  upon  each  annual  amount  of 
interest  (r)  upon  the  original  principal  of  £1.     The  table  is  as 

follows  :  — 

TABLE  YI,  B. 

Showing  the  amount  of  £1,  for  5  years  at  5  per  cent,  per  annum. 
Calculation  (lY)  1.  Showing  also  the  annuity  of  (r)  =  005, 
and  its  accumulations. 


1 

At 

end 

of 

year. 

2 

Amount 

of  £L 

5  years 

5%. 

3 

Annual 

Interest 

on  £1 

Principal. 

4                   5                  6 

Accumulation  of  (r)  Annu 
Interest  on  £1  at  end  of 

7 
al 

8 
Total 
Accumu- 
lation 

2nd 

3rd 

4tli 

5th 

of  (r). 

10000 

1 

■0500 

•0500 

•0025 

•0026 

•0028 

•0029 

•0108 

10500 

2 

0525 

•0500 

•0025 

•0026 

•0028 

•0079 

11025 

3 

•0551 
11576 

•0500 

— 

" 

•0025 

•0026 

•0051 

4 

•0579 

•0500 

" 

•0025 

•0025 

12155 

5 

•0608 

•0500 

— 

— 

— 

— 

— 

1-2763 

•2500 

•0025 

•0051 

•0079 

•0108 

•0263 

1^0000  Original  Sum 
•2500  Annual  Interest 
•0263  Accumulations  of  Annual  Interest 

~m63 


THE   AMOUNT   OF   ONE   POUND   PER   ANNUM  57 

The  original  principal  sum  of  £1  may  now  be  left  out  of 
the  calculation,  and  be  considered  only  as  the  origin  of  an 
annual  sum  or  annuity,  of  £005  to  be  accumulated  for  5  years 
at  5  per  cent,  per  annum  compound  interest.  It  is  in  fact,  at 
5  per  cent,  the  present  value  of  a  perpetual  annuity  of  £0'05. 
The  results  obtained  in  the  above  table  will  now  be  translated 
into  terms  of  the  formula  A  =  P  11^,  writing  against  each  factor 
in  the  arithmetical  result  the  corresponding  symbol  in  the 
formula,  but  as  the  formula  is  being  considered  in  its  relation  to 
£1  only,  there  will  be  substituted  for  P  its  equivalent  1,  with 
the   following   result,  viz.,    A  =  R^. 

The  above  Table  YI,  B,  expressing  the  results  of  Calcula- 
tion IV  (1)  may  be  analysed  as  follows:  — 

Actual 

results.    Formula. 

Amount  of  £1  in  5  years  at  5  per  cent.  ...     1'2763         R^ 
Deduct,   the    principal   sum   of*  which 
this  is  the  amount  at  the  end  of 
5  years  (Table  I)      l"  1 


leaving  ...     02763         EN-1 


which  is  the  accumulated 
amount  of  the  annvial  interest 
upon  £1  at  5  per  cent.,  or 
105-1  =  005      005  E-1 


which   is  the  annuity  which  will  in   5   years   at   5   per   cent., 
amount  to  £0"2763,  as  shown  in  the  above  table,  No.  YI,  B. 

The  formula  relating  to  the  accumulation  of  an  annual  sum 
is  derived  from  the  foregoing  results  as  follows  :  — 

It  has  been  ascertained  by  means  of  the  formula 
A  =  P  11^  relating  to  the  accumulation  of  £1 
as  given  in  Table  I,  that  the  amount  (A), 
of  £1  (P)  at  the  end  of  any  number  of  years  is     11^ 

and  by  deducting  therefrom  the  original  sum,  P, 

or  its  equivalent,  which  in  this  case  is 1 


a  constant  is  obtained   which  will   apply  to   any 

rate  per  cent.,   namely     11^  — 1 


58     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

This  constant  represents  the  accumulated  amount 
of  the  annual  interest  upon  £1,  resulting 
from  the  accumulation  of  the  original 
principal  sum  P,  at  the  ratio  R,  for  N  years. 

In  the  above  example,  the  ratio,  which  is  P,  plus 

one  year's  interest,  is       R  or  1"05 


and  by  deducting  therefrom  the  original  principal 

sum   P,    or   1.    the   remainder   is Il-lor0"05 


which  represents  the  interest  upon  £1  for  one  year 
and  is  constant  for  any  rate  per  cent. 

Expressing  the  above  in  terms  of  the  calculation  in  Table  VI,  B, 
it  is  found  that :  — 

(1{N_1)  or  0'2T6y  is  the  accumulated  amount  of  an  annual 
sum  of  (Ii-1)  or  0-05  for  (N)  or  5  years,  at  a  ratio  (R)  1-05, 
which  is  the  equivalent  of  5  per  cent,  per  annum. 

Stated  in  the  form  of  a  proportion  the  problem  becomes  :  — • 
If  0-05  per  annum  or  (R-1)  amounts  to  0-2763  or  (R^-l), 
what  sum  will  £1  per  annum  amount  to  under  the  same 
conditions,  as  follows  :  — 

0-2763  RN-1 

= =     5-526 

0-05  R-1 

which  agrees  with  the  amount  given  in  Table  III,  and  provides 
a  formula  which  may  be  used  to  calculate  the  amount  of  £1  per 
annum  for  any  number  of  years  at  any  rate  per  cent.  To  find 
the  amount  of  any  other  annual  sum  all  that  is  required  is  to 
multiply  the  result  obtained  in  the  above  manner  by  the  annual 
sum  in  question. 

It  is  not  possible  to  simplify  the  above  factor  (RN-  1)  because 
RN  varies  with  each  number  of  years,  but  (R-1)  may  be 
expressed  by  a  simple  symbol  because  it  is  always  constant 
for  eacli  rate  per  cent.  It  may  be  found  by  deducting  unity 
from  R  or  by  dividing  the  rate  per  cent,  by  100.  Tlie  factor 
(R-1)  is  denoted  by  the  symbol  (/■)  to  show  at  once  its  relation 
to,  and  variation  from,  the  factor  (R)  from  whicli  it  is  derived. 
The  amount  of  £1  per  annum  is  denoted  by  the  symbol  (M) 
to  distinguish  it  from  (A)  the  amount  of  £1. 


THE  AMOUNT   OF   ONE   POUND   PER   ANNUM  59 

The   formula   therefore   becomes  :  — 


(1)  as  to  £1  per  annum  : 


^r      /RN-1 

M: 


and 

(2)  as  to  any  annual  sum  (Ay)  : 


--^^C~^) 


and  the  symbols  have  the  meanings  described  in  Chapter  III. 

The  annuity  or  other  periodic  sum  in  all  cases,  as  already 
pointed  out,  is  presumed  to  be  paid  or  received,  set  aside  or 
invested,  at  the  end  of  the  first  and  each  succeeding  year,  which 
is  the  usual  method  in  all  annuity  calculations.  If  it  be  set 
aside  at  the  beginning  of  the  year  the  calculation  is  somewhat 
different. 

Calculations.  Having  found  the  above  formula  relating 
to  the  amount  of  £1  per  annum  in  any  number  of  years,  two 
calculations  will  now  be  made  by  its  aid,  upon  the  author's 
standard  form  Xo.  o.  Both  will  include  the  three  methods  of 
which  the  general  rules  are  stated  at  the  head  of  this  chapter, 
namely,  by  formula,  by  the  published  tables,  and  by  Thoman's 
method  and  tables.  The  first  calculation  will  deal  only  with 
an  annuity  of  £1,  and  will  show  the  method  of  computing  the 
amounts  given  in  Table  III,  Calciilation  (VI)  1.  The  second 
calculation  will  deal  with  an  annuity  of  stated  amount,  and 
will  illustrate  the  method  to  be  adopted  in  actual  practice. 
Calculation  (VI)  2. 


6o 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Calculation  (VI)  1. 

Standard  Calculation  Form,  No.  3. 

To  find  the  amount  of  an  annuity  in  any  number  of  years,  and 

thereby  prove  the  accuracy  of  the  published  table. 

Table  III. 
Required  the  amount  of  £1  per  annum  for  10  years  at  5  per  cent. 

per  annum  compound  interest. 

(A)     By  Formula.  ^i  =  Ky  (^^^^^  ^^^1©  1,  Chapter  YI. 


Log. 
R^— 1^ 


Log.  Ratio  I  R 

multiply  Log.  R  by      N 


Convert    Log.    to 

ordinary  number 
deduct  unity 

Log.  of  this  is 

Log.  Annuity 
add.  Log .  RN  - 1  above 


deduct  Log.  r 


105 
10 


00211893 
10 


n^ 

(l-05jio 

0-2118930 

RN 

-1 

1-6289 
1- 

RN-l 

0-6289 

1-7985779 

A:y 

RN-l 

1- 

00000000 
1-7985779 

A2/(RN- 
r 

-1) 
•05 

1-7985779 
2-6989700 

M 

1-0996079 

Required  future  amount,  £12-5779. 


(B)     By  Table  III.  M  =  A.y  (^    —^\  Rule  2,  Chapter  YI 


Table  III.     10  years,  5  per  cent. 
Amount  of  £1  per  annum 
Add  Log.  Annuity 


RN-l 


Ky 


12-5779 


M 


Required  future  amount,  £12-5779.     This  amount  is  given  in 
Table  III. 

(C)     By  Thoman's  Table.      M  =  Ky  (^  \    Rule  3,  Chapter  YI. 

5  per  cent.  10  years. 

Log  Annuity 
Add  Log.  RN  in 

Table +10 


deduct  Log.  a" 


Required  future  amount,  £12- 57 79. 


A,y              1- 

RN 

0-0000000 
10-2118930 

A2/RN 

10-2118930 
9-1122851 

M 

1-0996079 

THE   AMOUNT   OF   ONE   POUND   PER    ANNUM 


Calculation  (VI)  2. 

Standard  Calculation  Form,  No.  3. 

To  find  the  amount  of  an  annuity  in  any  number  of  years. 

Table  III. 
Required  the  amount  of    £500  per  annum  for  10  years  at  5  per 
cent,   per  annum  compound  interest. 

(A)     By  Formula .  M  =  Ai/   ("  ~-^  )  I^^il©  1 ,  Chapter  VI . 


'  Log.  Ratio 

viultiply  Log.  R  by 

Log.    I 
RN  -I"*   Convert  Log.  to 

ordinary  number 
deduct  unity 

I       Log.  of  this  is 

Log  Annuity 
add  Log .  RN  -  1  above 


deduct  Loff.  r 


1-05 
10 


00211893 
10 


RN 

(ro5)i« 

0-2118930 

RN 

-1 

1-6289 
1- 

RN-1 

0-6289 

T- 7985779 

Ay 

RN-1 

500 

2-6989700 
1-7985779 

Ai/(RN- 
/■ 

-1) 

2-4975479 
2-6989700 

M 

3-7985779 

Required  future  amount,  £6288-94 


(B)     By  Table  III.  U  =  Ay  (^^^ — ^  Rule  2,  Chapter  YI 


Table  III.     10  years,  5  per  cent. 
Amount  of  £1  per  annum 
add  Log.  Annuity 


RN-1 

12-5779 
500 

10996079 

A.y 

2-6989700 

M 

3-7985779 

Required  future  amount,  £6288-94 


(C)     By  Thoman's  Table.      M^At/ ("^^    Rule  3, 
5  per  cent.  10  years. 


Chapter  VI. 


Log  Annuity 
Add  Log.  RN  in 

Table +10 

Ay 

RN 

500 

2-6989700 
10-2118930 

deduct  Log.  a" 

Ai/RN 

a^ 

12-9108630 
91122851 

M 

3-7985779 

Required  future  amount,  £6288-94 


62  REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


CHAPTEH  VII. 

COMPOUND  INTEREST  AS  APPLIED  TO  AN  ANNUAL 
OR   OTHER   PERIODIC  PAYMENT   {Cojitinued). 

TABLE  IV.     The  present  value  of  £i  per  annum  for  any 
number  of  years. 


-%(^) 


FORMUL.^  USED  IN  CALCULATIONS  AND  RULES  DEDUCED 
THEREFROM.  DERIVATION  OF  FORMULA  AND  APPLICATION  TO 
COMPILATION  OF  TABLES  AND  TO  CALCULATIONS.  CALCULATIONS 
TO  DEMONSTRATE  THE  THEORETICAL  CONCLUSIONS  BOTH  AS 
REGARDS  THE  PUBLISHED  TABLES  AND  PRACTICAL  EXAMPLES. 
ThOMAN's  method   and   FORMULA. 

Author's  Standard  Calculation  Form,  No.  4. 


Formulae. 

A.   To   find   file   present    vahie   of  £1   per   annum  for   any 
ninnher  of  years,  as  given  in  the  puhlished  tables  :  — 

(7)  Formula,  P=   (^-jpr^^ 

by   logs.:      Log.    [Freserit    value   of  £1   per  annum)  = 
Log.  {W^- 1) -Log.  RN-Log.  r. 

{2)  By  Th Oman's  'method:  — 

Formula,  P=  -— 

a" 

by  logs.  :      Log.  {Present  value  of  £1  per  annum)  — 

10 -Log.  rt'i 

B.  To  fnd  the  present  value  of  any  annuity  for  any  number 

of  years  :  — 

/RN_1\ 

(7)  Formula,  V  =  Ky\^^^^^  ^ 

by  logs.  :      Log.     [Present     value     of     annuity)  =  Log. 
Annuity +  Log.  {ll^-l)-Lug.  W^-Log.  r 

[2]   By  Tho)na7i's  method:  — 

Formula,  P=  —;— 

a'' 

by  logs.:      Log,    [Present   value   of   annuity)  =  Log . 
annuity  +  10  —  Log.  a"- 


THE  PRESENT  VALUE  OF  ONE  POUND  PER  ANNUM      63 
The     'present     chapter     deals     only     with     the     formula 

(T>  N 1  X 
—  y     Th Oman's    method    and   formula'    are    fidly 

described  in  Chapter  IX. 

General  Rules  deduced  from  the  above  formulae. 

To  find  the  present  value  of  any  annuity  for  any  number  of 
years.  Author's  Standard,  Calc\iJation  Form,  No.  4. 

Rule  1.     If  the  rate  per  cent,  be  not  given  in  Table  IV  or  in 
Thoman's  Tables:  — 

Proceed  by  the  formula  relating  to  Table  IV. 

Calculation  {VII)  2  A. 

Rule  2.     If  the  rate  per  cent,  be  given  in  Table  IV  : — ■ 

Multiply  the  amount  given  in  the  table,  by  the 
given  annuity.  The  product  is  the  present  value 
required.  Calculation  {VII)  2  B. 

Rule  3.     If  the  rate  per  cent,  be  given  in  Thomans  Table  :  — 
To  the  log.  of  the  given  annuity  add  10,  and  deduct 
therefrom  the  log.  of  a"-  as  given  by  Thoman.     The 
remainder  is  the  log.  of  the  present  value  required. 

Calculation  {VII)  2C. 

To  fnd  the  rate  per  cent.,  or  number  of  years,  proceed  as 
shown  in  the  standard  form  for  the  purpose,  given  in 
Chapter  X. 


Derivation  of  the  Formula.  In  order  to  find  tlie  formula 
relating  to  the  present  value  of  an  annuity  due  at  the  end  of 
each  year  of  a  given  term,  the  most  direct  method  is  to  consider 
an  annuity  of  £1  in  order  to  demonstrate  the  principle  involved 
and  to  arrive  at  the  necessary  modification  in  the  previous 
formula  relating  to  the  amount  of  an  annuity.  It  will  be 
readily  seen  that  the  present  value  of  an  annuity  for  any 
number  of  years  is  the  same  as  the  present  value  of  the  sum  to 
which  that  annviity  will  amount  in  the  same  period  at  the  same 
rate  per  cent.  It  has  been  shown  in  Chapter  \1  that  the 
amount  of  an  annuity  of  £1  may  be  found  by  the  formula  :  — 

r 
and  that  the  present  value  of  £1  due  at  the  end  of  any  number 


64  REPAYMENT    OF    LOCAL   AND    OTHER   LOANS 

of  years  is  found  by  tlie  formula,  relating  to  Table  II  and 
described  in  Chapter  Y,  namely :  — 

p=A 

but  A  —  1 ,  therefore  P  =  ^j^^ . 

Consequently  by  multiplying  these  two  formulae  together 
the  required  formula  for  the  present  purpose  is  obtained  as 
follows  :  — - 

R^  - 1        1        R^  —  1 

P  =  (present  value  of  £1  per  annum)  =  ^  "dn  =  ""^dn — 

and  the  formula  to  find  the  present  value  of  any  annuity,  Ky, 
for  any  number  of  years  becomes:  — 

'   .     . .  '-M^) 

There  is  a  similarity  between  the  formuli3e  relating  to 
Tables  III  and  IV,  namely,  that  they  are  both  based  upon  the 

RN  - 1 

factor  ; —  .     In  both  cases,  as  will  be  seen  by  an  inspection 

of  the  standard  calculation  forms.  Rule  1,  the  method  consists 
in  adding  to  the  log.  of  the  annuity  the  log.  of  R^-l^  and 
deducting  from  the  sum  of  the  logs,  the  log.  of  r.  This  gives 
the  desired  result  in  the  case  of  Table  III  relating  to  the 
amount  of  an  annuity,  but  in  Table  IV  relating  to  the  present 
value  of  an  annuity,  the  log.  of  R^  is  previously  deducted  from 
the  log.  of  the  annuity,  which  is  ecjuivalent  to  saying  that 
the  present  values  in  Table  IV  may  be  found  by  dividing  the 

amounts  in  Table  III  by  R^ ;  but  the  values  of  -jj^  are  given  in 

Table  II,  therefore  the  amounts  in  Table  IV  are  equal  to  the 
amounts  in  Table  III,  multiplied  by  the  amounts  in  Table  II, 
or  divided  by  the  amounts  in  Table  I. 

Calculations.  Having  found  the  above  formula  relating 
to  the  present  value  of  an  annuity  of  £1  for  any  number  of 
years,  two  calculations  will  now  be  made  by  its  aid  upon  the 
author's  standard  calculation  form,  No.  4,  Both  cases  will 
include  the  tliree  methods  of  which  the  general  rules  are  stated 
at  the  head  of  this  chapter,  namely,  by  formula,  by  the 
published  tables,  and  by  Thoman's  method  and  tables.  The 
first  calculation  will  deal  only  with  an  annuity  of  £1,  and  will 
show  the  method  of  computing  tlie  amounts  given  in  Table  IV. 
Calculation  (VII)  1. 

The  second  calculation  will  deal  with  an  annuity  of  stated 
amount,  and  will  illustrate  the  method  to  be  adopted  in  actual 
practice.     Calculation   (VII)  2. 


THE  PRESENT  VALUE  OF  ONE  POUND  PER  ANNUM      65 


Calculation  (VII)  1. 

Standard  Calculation  Form,  No.  4. 

To  find  the  present  value  of  an  annuity  for  any  number  of  years, 

and  thereby  prove  the  accuracy  of  the  published    table. 

Table   ly. 
Required  the  present  value  of  £1  per  annum  for  10  years  at 
5  per  cent,  per  annum,  compound  interest. 

(A)     By  Formula.  P  =  A2/('^j^)   Rule  1,  Chapter  VII. 


'  Log.  Ratio 

multiply  Log.  R  by 

Convert  Log. 

to  ordinary  number 
deduct  unity 

^       Log.  of  this  is 
Log.  Annuity 
add  Log.  (RN  - 1)  above 

deduct  Log.  R^  above 
deduct  Log.  r 

R 

N 

RN 

1-05 
10 

(1-05)10 

0-0211893 
0-2118930 

Log     . 
W  -1 

RN 

-1 

1-6289 
1- 

RN-1 

0-6289 

1-7985779 

Ay 

RN-1 

1- 

0-0000000 
1-7985779 

RN 

1-7985779 
0-2118930 

r 

1-5866849 
2-6989700 

P 

0-8877149 

Required  present  value,  £7-72174. 


(B)     By  Table  lY.  P  =  A.y  (^^£r^)  ^^^1©  2,  Chapter  VII. 


Table  IV.     10  years,  5  per  cent. 
Present  Value  £1  per  annum 
add  Log.  Annuity 


RN-1 


RNr 
A.y 


7-72174 


Required  present  value,  £7-72174.     This  amount  is  given  in 

Table  IV. 


(C)      By  Thoman's  Table 


Ky 


Rule  3,  Chapter  VII. 


5  per  cent.  10  years. 

Log.  Annuity 

A.y 

1- 

0- 

a^^lO 
deduct  Log. 

a« 

100000000 
9-1122851 

P 

0-8877149 

Required  present  value,  £7-72174. 


66 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Calculation  (VII)  2. 

Standard  Calculation  Form.,  No.   4. 

To  find  tlie  present  value  of  an  annuity  for  any  number  of  years. 

Table   IV. 
Required  the  present  value  of  £500  per  annum  for  10  years 
at  5  per  cent,  per  annum,  compound  interest. 

(A)     By  Formula.  P  =  Ay  (^^n^)  R"le  1,  Chapter  YII. 


Log 
RN_i 


^  Log.  Ratio  R  105 

nniltiply  Log.  R  by     N  10 


-1  Convert  Log, 

to  ordinary  number 
deduct  unity 

Log-,  of  this  is 


Log.  Annuity 

add  Log.  (RN  - 1)  above  RN  _  l 


deduct  Log.  R^  above  RN 


deduct  Log.  r 


Required  present  value,  £8860-86" 


0021189:3 
10 


RN 

(1-05)10 

0-2118930 

RN 

-1 

1-6289 
1- 

RN_1 

0-0289 

r7985779 

Ay 
RN-l 

500 

2-6989700 
1-7985779 

RN 

2-4975479 
0-2118930 

r 

2-2856549 
2-6989700 

P 

3-5866849 

(B )     By  Table  lY.  P  =  Ay  /"^^^^^  Rule  2,  Chapter  YII 


Ti 

iblelY.     10  years, 

Present  Yalue  £1 

Add  Log 

5  per  cent, 
per  annum 
Annuity 

RN- 

-1 

7-72r 

500 

'4 

0-8877149 
2-6989700 

3-5866849 

RN; 

Ay 

P 

Required  present  value,  £;)860-867 


(C)     By  Thoman's  Table.  P: 

5  per  cent.  10  years. 


Ay 
a" 


Rule  3,  Chapter  YII. 


Log.  Annuity 

Ay 

500 

2-6989700 

add  10 
deduct  Log. 

a"" 

12-6989700 
9-1122851 

P 

3-5866849 

Required  present  value,  £3860867. 


THE    ANNUITY    ONE    POUND    WILL    PURCHAvSE  67 


CHAPTER  Till. 

COMPOUN'D  INTEREST  AS  APPLIED  TO  AN  ANNUAL 
OE  OTHER  PERIODIC  PAYMENT  {Contimied). 

TABLE  V.     The  annuity  which  £1  will   purchase  for  any 
number  of  years,  or  of  which  £1  is  the  present  value. 


--^  (^d 


FOEMUL.E     AND      RULES      DEDUCED      THEEEFROM.  GENERAL 

REMARKS    AS    TO    TaBLE    V    AND    ITS    RELATION    TO    AN    EQUAL 

annual  instalment,  of  i'rincip.al  and  interest  combined. 
This  table  gives  the  actual  values  of  Thoman's  log. 
FACTOR,  a".  Derivation  of  formula  and  application  to 
compilation  of  tables  and  to  calculations.  Calculations 
to  demonstrate  the  theoretical  conclusions  both  as 
regards  the  published  tables  and  practical  examples. 
Thoman's  method  and  formula. 

Author's  Standard  Calculation  Form,  No.  5. 


Formulae. 

A.  To  find  the  annuity  ivhich  £1  will  imrchase  for  any 
number  of  years,  or  of  wliich  £1  is  the  ijresent  value,  as 
given  in  the  inihltshed  tables  :  — 

(1)  Formula,  Ai/=  (  ]p"^  J 

by  logs.:      Log.  (Annuity  £1  will  i)urchase)  =  Log .  R^ 
+  Log.T-Log.(W^-V) 

[2)  By  Thoman's  method:  — 

Formula,  Ay  =  a^ 

by  logs.:      Log.  (Annuity  £1  icill  pxir chase)  =  Log. 
a«-10 


68     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

B.  To  fnd  the  annuity  ivhich  may  be  ijurcliased  with  any 
given  sum  for  any  number  of  years  :  — 

(1)  Formula,  Ay  =  V  (jpr^l) 

by    logs.  :      Log.    [required   annuity)  =  Log.    {jjrincifal 
sum)  +  Log.  W^  +  Log.  r- Log.  (RN-l) 

(2)  By  Tho man's  method  :  — 

Formula,  Ay  =  V  a" 

by  logs.  :      Log.  (required  annuity)  =  Log.  [yrinci'pal 
sum)  +  Log.  «"•  — 10 

The  present  chapter  deals  only  ivith  the  formula 

Thoman's  method  and  formulce  are  fully  described  in  Chapter 
IX. 

General  Rules  deduced  from  the  above  formulae. 

To  find  the  annuity  which  may  be  purchased  with  any  given 
sum  for  any  number  of  years. 

Author's  Standard  Calculation  Form,  No.  5. 

Rule  1.     If  the  rate  per  cent,  be  not  given  in  Table  Y  or  in 
Thoman's  Tables:- — ■ 

Proceed  by  the  formula  relating  to  Table  V . 

Calculation  {VIII)  2  A. 

Rule  2.     If  the  rate  per  cent,  be  given  in  Table  V  :  — 

Multiply  the  annuity  given  in  the  table,  by  the  given 
sum.  The  product  is  the  reqtiired  annuity  which 
may   be   purchased.  Calculation  [VIII)  2  B. 

Rule  3.     If  tJie  rate  per  cent,  be  given  in  Thoman's  Tables  :  — 
To  the  log.  of  the  given  sum,  add  the  log.  of  a"  as 
given  by  Thoman.     Deduct  10  from  the  sum  of  the 
two  logs.       The  remainder  is  the  log.  of  the  required 
annuity  which  may  be  purchased. 

Calculation  (VIII)  2  C. 

To  find  the  rate  per  cent.,  or  number  of  years,  proceed  as 
shown  in  the  standard  form  for  the  purpose^  given  in 
Chapter  X. 


Tables  III  and  IV,  containing  the  amounts  and  present 
values  of  £1  per  annum,  correspond  to  Tables  I  and  II  relating 
to  tbe  amounts  and  present  values  of  £1.     Tliere  is  a  further 


THE    ANNUITY    ONE    POUND    WILL    PURCHASE  69 

Table,  No.  Y,  given  in  Inwood  and  other  published  tables, 
which  is  useful  in  order  to  ascertain  the  annuity  which  may  be 
purchased  with  a  given  sum  of  money,  because  anyone 
contemplating  the  purchase  of  an  annuity  generally  has  a 
definite  sum  to  invest  in  this  manner.  Consequently  it  is 
required  to  know  the  annuity  which  £1  will  purchase  for  any 
number  of  years,  and  from  this  can  be  ascertained  by  simple 
multiplication  the  annuity  which  any  given  sum  will  purchase. 
But  the  principal  value  of  this  table  lies  in  the  fact  that 
the  amounts  there  given  represent  the  respective  annuities  of 
which  £1  is  the  present  value.  The  importance  of  this  will  be 
recognised  when  it  is  remembered  that  this  is  the  principle 
underlying  the  repayment  of  debt  by  an  equal  annual  instal- 
ment of  principal  and  interest  combined,  as  laid  down  in 
Section  234(4)  of  the  Public  Health  Act,  1875.  Table  Y 
represents  Thoman's  factor  {a"'),  and  is  a  connecting  link 
between  £1  and  £1  per  annum  considered  both  in  regard  to 
future  amount  and  present  value.  By  its  aid  the  cumbersome 
factor  E^-l,  previously  referred  to,  may  be  avoided  in  cases 
where  the  rate  per  cent,  is  included  in  Thoman's  tables. 

Deeivatiox  of  the  Foemfla.  The  formula  relating  to  this 
table  may  be  found  by  simple  proportion  without  resorting  to 
any  algebraical  calculation.  It  has  been  ascertained  by 
Calculation  (YII)  1,  that  £7-7217  is  the  present  value  of  an 
annuity  of  £1,  and  such  values  are  given  in  Table  lY.  It  is 
required  to  find  the  annuity  of  which  £1  is  the  present  value. 

It  is  obvious  that  it  will  be    ^  „,.-,^   of  £1. 

<  w21< 

Consequently,   by  dividing  unity  by  the  present  value  of  an 

annuity  of  £1,  as  given  in  Table  lY,  the  result  is  the  annuity 

which  may  be  purchased  by  £1,  and  may  be  expressed  by  the 

following  rule  :  — 

To  find  the  annuity  which  £1  will  purchase  for  any  number 
of  years,  first  ascertain  by  Table  IV  the  present  value 
of  an  annuity  of  £1  for  the  same  period  at  the  same 
rate  per  cent.;  and  divide  1  by  the  present  value  so 
found.  The  quotient  tvill  be  the  annuity  ivhich  may  be 
purchased  by  £1. 

This  rule  simply  means  that  to  ascertain  the  annuity  which 
£1   will   purchase,   unity    is   divided   by   the  values    given    in 


70  REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 

Table  1\,  but  if  it  be  reduced  to  terms  of  tlie  annuity  formula 

it  becomes  :  — 

^  1  R^  r 

The  annuity  of  which  £1  is  tbe  present      [  ^^TTTi    *^^'   j^n  _  x 

value,  as  given  in  Table  Y I  p jj 

Table  Y.  Calculations.  Having  found  the  above  formula 
relating  to  the  annuity  which  £1  will  purchase  for  any  number 
of  years  or  the  annuity  of  which  £1  is  the  present  value,  two 
calculations  will  now  be  made  by  its  aid  upon  the  author's 
standard  form,  No.  5.  Both  cases  will  include  the  three 
methods  of  which  the  general  rules  are  stated  at  the  head  of  this 
chapter,  namely,  by  formula,  by  the  published  tables,  and  by 
Thoman's  method  and  tables.  The  first  calculation  will  deal 
only  with  the  annuity  which  £1  will  purchase,  or  of  which  £1 
is  the  present  value,  and  will  show  the  method  of  computing  the 
amounts  given  in  Table  Y.     Calculation  (YIII)  1. 

The  second  calculation  will  deal  with  a  stated  amount  to  be 
invested  in  an  annuity,  or  of  Avhieh  it  is  required  to  ascertain 
the  future  equivalent  expressed  in  an  annual  payment,  and 
will  illustrate  the  method  to  be  adopted  in  actual  practice. 
Calculation  (YIII)  2. 


THE    ANNUITY    ONE    POUND    WILL    PURCHAvSE 


71 


Calculation  (VIII)  1. 

Statulard  Calculation  Form,  No.  5. 

To  find  the  annuity  which  a  prevsent  sum  will  purchase  for  any 
number  of  years,  and  also  the  equal  annual  instalment  of 
principal  and  interest  combined,  and  thereby  to  prove  the 
accuracy  of  the  published  table.  Table  Y. 

Eequired  the  annuity  Avliich  £1  will  purchase  for  10  years  at 
5  per  cent,  per  annum,  compound  interest. 

(A)     By  Formula.  A//  =  P  ( j^^-^j)  I^^^lt'  1,  Chapter  Till. 

Log'.  Ratio 

)nu.ltiply  Log.  R  by 


Jog    .    Convert  IjO^. 


W  -1 


to  ordinary  number 
deduct  unity 


Log.  Present  Sum 
add  Log.  RN  above 
Loff.  r 


deduct  Log.  (R^  -  1)  above 


R 

105 
10 

0-0211893 
10 

RN 

(1-05)10 

0-2118930 

RN 
-1 

1-6289 
1- 

RN-1 

0-6289 

i- 7985779 

P 
RN 

/' 

1- 

1-6289 
0-05 

0- 

0-2118930 
2-6989700 

RN-1 

2-9108630 
1-7985779 

Ay 


1-1122851 


Required  annuity,  £0-129546. 


(B)     By  Table  Y.  Ay  =  V  (j^^^)  Rule  2,  Chapter  YIII 


Table  Y.     10  years,  5  per  cent. 
Annuity  £1  will  purchase 
add  Loo'.  Present  Sum 


RNr 


RN 
P 


0-1295 


A) 


Required  annuity,  £0-1295.     Tliis  amount  is  given  in  Table  Y. 


(C)     By  Thoman's  Table.         A.y  =  P  r/"       Rule  3,  Chapter  YIII. 
5  per  cent.  10  years. 

Log.  Present  Sum 
add  Los".  «"■ 


deduct  10 


IP 

I  an 

1- 

0- 

91122851 

9-1122851 

A;?/ 

11122851 

Required  annuity,  £0-129546. 


72 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Calculation  (VIII)  2. 
Standard  Calculation  Form,  No.  5. 
To  find  tlie  annuity  which  a  present  sum  Avill  purchase  for  any 

number  of  years.  Tabled. 

To  find  the  annuity  which  may  be  purchased  with  £6288-94  for 

10  years  at  5  cent,  per  annum,  compound  interest. 

(A)     By  Formula  A2/  =  P  (f^-3^1)  I^^^le  1,  Chapter  YIII. 


Log.  Ratio 

multiijly  Log.  R  by 

Convert  Log. 

to  ordinary  number 
deduct  unity 

R 

N 

105 
10 

0-0211893 
10 

RN 

(1-05)10 

0-2118930 

R^^  -1 

RN            1-6289 
-1         1- 

RN- 

-1       0-6289 

1-7985779 

Log.  Present  Sum 
add  Log.  RN  above 
Log.  r 

P 

RN 

r 

6288-94 
1-6289 
0-05 

3-7985779 
0-2118930 
2-6989700 

deduct  Log.  (R^  - 1)  above 

RN- 

-1       0-6289 

2-7094409 
1-7985779 

Ay 

2-9108630 

Required  annuity,  £814-447 


(B)     By  Table  Y.  Ay^V  (j^^;,)  Rule  2,  Chapter  YIII. 


Table  Y.     10  years,  5  per  cent. 
Annuity  £1  will  purchase 
add  Log.  Present  Sum 


RNr 


RN-1 


01295 

6288-94 


1-1122851 
3-7985779 


Ay 


2-9108630 


Required  annuity,  £814-447 


(C)     By  Thoman's  Table.         A.y  =  P  a«       Rule  3,  Chapter  YIII. 
5  per  cent.  10  years. 


Log.  Present  Sum 
add  Log.  a" 

P 

a"- 

6288-94 

3-7985779 
9-1122851 

12-9108630 

deduct  10 

Ay                                 2-9108630 

Required  annuity,  £814-447. 


THOMAN'S    TABLES  73 


CHAPTER  IX. 

THOMAN'S  LOGARITHMIC  TABLES  OF  COMPOUND 
INTEREST  AND  ANNUITIES. 

Explanation  of  Thoman's  symbols,  R^,  and,   a",  and  their 

RELATION,  SEPARATELY  OR  IN  COMBINATION,  TO  THE  FORMULA 
ALREADY  ASCERTAINED.       ThOMAN's  METHOD  OF  STATING   LOG. 

of  a"-  by  adding  10  to  the  log. 

Author's  Standard  Calculation  Forms,  1  to  5. 


Symbols  used  by  Thoman  : — 

'R^=the  amount  of  £1  in  any  number  of  years. 
a"'  =  the  annuity  lohicli  £1  ivill  'purchase  for  any  number 
of  years. 


an  = 


Rn  _  1  [Table   V. 


Comparison  with  previous  formulae  :  - 

Thoman's 
Table  General    Logarithmic 

Chapter.    No.  Giving  Values  of  Formulas.    Formulse. 

IV.       1.       Amount  of  £1       R^        R"" 

F.         II.     Present  value  of  £1     ^on       ""dn 


VI.       III.  Amount  of  £1  per  annum 


RN  -1      RN 

r  a"- 


RN  -  1      1 

VII.  IV.    Present  value  of  £1  per  annum  ...       -^^ —     -^ 

RNr 

VIII.  V .      Annuity  which  £1  will  purchase       ^.^      -       t'-" 

The  logarithmic  equivalents  of  the  above  formulae  and  the 
rules  based  thereon  are  given  at  the  head  of  the  chapter  dealing 
with  each  table. 


74     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Inchided  iu  Inwood,  and  iu  other  published  tables  of 
compound  interest,  are  a  series  of  valuable  tables  by  M,  Fedor 
Thoman,  of  the  Soc.  Credit  Mobilier,  of  Paris,  the  author  of 
"  Logarithmic  Tables  of  Interest,  etc."  These  tables  are  of 
great  assistance  in  the  solution  of  many  problems;  and  the 
whole  of  the  formulae  already  obtained  by  derivation  from  the 
algebraical  formula  for  a  geometrical  progression  will  now  be 
compared  with  the  simplified  formulse  given  by  Thoman.  A 
glance  at  the  above  comparative  table  will  show  that  the  whole 
of  the  formula  already  obtained  by  derivation  from  the  formula 
A  =  P  E^,  may  be  expressed  by  some  modification  of  the  factors 
E^',  and  /■,  which  have  already  been  fully  explained  iu  previous 
chapters.  The  table  at  the  end  of  Chapter  Y  contains  the 
values  of  E,  and  /•,  for  many  rates  per  cent,  likely  to  be 
required  in  practice,  and  also  gives  the  corresponding  logs,  of 
these  values ;  and  it  has  been  explained  how  to  find  by  means 
of  the  formulse  the  values  and  logs,  of  E,  and  /•,  for  any 
intermediate  rates  per  cent,  not  inchided  in  the  above  table. 
The  values  of  E^'  may  be  obtained  by  multiplying  the  log.  of  E 
by  the  number  of  years,  as  shown  in  Calculation  (IT)  2  and 
others. 

Problems  involving  an  annual  or  other  periodic  payment,  as 
in  Tables  III,  lY,  and  Y,  cause  the  introduction  of  a  variation 
of  E^',  namely  (E^^-1),  which  imports  a  new  calculation  which, 
although  not  of  itself  difficult,  is  inconvenient  because  it  is 
necessary  to  convert  the  log.  of  E^\  to  an  ordinary  number 
before  deducting  unity,  and  afterwards  to  find  the  log.  of  the 
remainder.  This  might  be  avoided  by  preparing  tables  of 
(E^-l),  and  the  corresponding  logs.,  for  each  rate  per  cent,  for 
any  number  of  years. 

In  the  above  formulse  relating  to  an  annual  sum  in  Tables 
•  III,  lY,  and  Y,  the  factor  (EN-1)  is  always  associated  with  i\ 
or  (E^^  ;•),  and  E^  is  the  factor  relating  to  the  amount  and 
present  value  of  £1  as  shown  by  Tables  I  and  II.  By 
combining  the  factors  (E^-l),  and  E^,  a  connecting  link  is 
obtained  between  Tables  I  and  II  relating  to  £1,  and  Tables 
III,  lY,  and  Y,  relating  to  the  amount  and  present  value  of 
£1  per  annum ;  and  this  is  the  principle  underlying  Thoman's 
method . 

They  are  merely  tables,  and  do  not  enunciate  any  new 
principle,  but  by  giving  under  each  rate  per  cent,  for  various 
numbers  of  years  the  logs,  of  two  factors  only,  they  enable  any 
calculation  to  be  made,  at  the  rates  included  in  the  tables, 
without  any  further  reference  to  the  ordinary  published  tables 


THOMAN'S    TABLES  75 

I  to  Y.  Tliomau's  tables  have  two  advantages  over  the  ordinary 
tables  of  compound  interest  in  that  (1)  they  are  worked  out  for 
fractional  eighths  per  cent,  up  to  6  per  cent.,  and  (2)  they  give 
the  logs,  direct  and  thereby  avoid  any  reference  to  the  log. 
tables ;  but  since  they  are  worked  out  for  a  limited  number  only 
of  rates  per  cent,  they  do  not  dispense  entirely  with  the  method 
of  calculation  by  means  of  the  formulae  previously  stated,  and 
these  methods  will  therefore  be  included  in  subsequent  chapters 
as  well  as  Thoman's  method. 

The  factors  included  in  Thoman's  tables  are  R^  and  a". 
EN  is  the  factor  governing  Tables  I  and  II,  Avithout  any 
alteration,  and  Thoman's  tables  may  be  referred  to  instead  of 
finding  the  logs,  of  these  values  by  the  methods  shown  in  the 
various  calculations,  using  the  ordinary  tables  and  logs. 

a«  is  used  by  Thoman  to  denote  the  annuity  which  £1  will 
purchase  or  of  which  £1  is  the  present  value. 

The  logs,  of  a^  in  Thoman's  tables  are,  purely  for  convenience 
of  calculation  and  perhaps  for  facilitating  printing,  given  in  a 
different  form  to  the  logs,  of  E^  in  the  same  tables.  In  Table  I  (E^) 
the  values  are  all  greater  than  unity,  hence  the  characteristics 
of  the  logs,  of  these  values  are  always  positive.  In  Table  V 
the  values  are  all  decimals  of  unity,  and  the  logs,  of  these 
values  have  negative  characteristics.  Thoman  adds  10  to  the 
characteristics  of  the  logs,  of  the  values  of  a''  in  Table  Y,  and 
thus,  bearing  in  mind  that  any  calculation  can  be  made  by 
means  of  E^  and  a'\  it  is  possible  to  eliminate  the  troublesome 
negative  characteristic  altogether.  All  that  is  required  is  to 
correct  the  characteristic  of  the  final  log.  by  adding  10  in  the 
case  of  Tables  III  and  lY,  and  deducting  10  in  the  case  of 
Table  Y,  to  or  from  the  resulting  log.  before  ascertaining  the 
antilog.,  or  numerical  equivalent.  Thoman's  logs,  of  a«  may  be 
treated  in  the  ordinary  way,  by  using  the  mantissa  given  in  the 
table  and  converting  the  characteristic  there  giA-en  to  the  proper 
minus  quantity,  i.e.,  10  minus  the  given  characteristic.  For 
the  sake  of  clearness  the  method  of  deducting  or  adding  10 
from  or  to  the  log.  has  been  adopted  in  the  whole  of  the 
standard  calculation  forms  prepared  by  the  author. 

There  are  two  methods  of  connecting  the  ordinary  tables  and 
formula-  relating  to  £1,  and  £1  per  annum,  namely,  either  by 
means  of  Table  I  and  III,  dealing  with  the  respective  amounts 
(as  adopted  by  Thoman  to  derive  the  factor  ««  as  shown  by 
Table  Y)  or  by  means  of  Tables  II  and  III,  leading  to  Table  lY, 
Avhich  is  the  reciprocal  of  Table  Y,  as  shoAvn  when  dealing  with 
the  latter  table  in  Chapter  YIII. 


76  REPAYIvrENT   OF   LOCAL   AND    OTHER   LOANS 

Thoman's  method  Avill  now  be  applied  in  order  to  derive 
Table  Y,  or  tbe  formula  relating  thereto,  from:  — 

Table  I,  .   the  amount  of  £1  for  any  number  of  years, 

and,  Table  III,  the  amount  of  £1  per  annum  in  any  number  of 

years, 
taking,    in   each    case,    a    period    of    10   years,    and    a    rate   of 
accumulation  of  5  per  cent,  per  annum,  as  follows:  — 

Table  I,      amount  of  £1       1-6289  or  RN 

Table  777,  amount  of  £1  per  annum  ...   12"5779  or  

The  annuity  which  £1  will  purchase  is  obtained  by  dividing 
16289  by  12'5TT9,  and  the  formula  corresponding  thereto  may 
also  be  obtained  by  dividing  the  corresponding  formulae  as 
follows  :  — 

Table  I         1-6289  _     R^ 
lable  V    -.rj^,.^blein     12-5779      W  -I' 

r 
Stated  in  actual  values  :  — 

1-6289 
Table  V  =  ,^":^^  =  0-1295 
125/ 79 

as  may  be  found  by  actual  calculation,  or  obtained  by  direct 
reference  to  Table  Y. 

Stated  in  terms  of  the  above  formulae  :  — 

Table  V  =  rra ^   or 


RN  -1        R^  -1 
I' 

which  is  the  formula  relating  to  Table  Y,  as  shown  in 
Chapter  YIII,  giving  the  annuity  which  £1  will  purchase 
for  any  number  of  years.  But  Thoman's  symbol  a»,  although 
expressed  in  logarithmic  form,  represents  the  same  factor; 
therefore  the  formula  relating  to  Table  Y,  as  derived  in  Chapter 
YIII,  from  the  simple  formula  A  =  P  RN,  may  be  replaced  by 
Thoman's  symbol  «",  with  the  result  that :  — 

Table  V  =^^  _x=^^"  «^"  Thoman. 

The  above  formula,  relating  to  Table  Y,  contains  the  three 
factors,  RN,  r,  and  (RN-1)^  already  referred  to  and  fully 
explained  in  previous  chapters,  in  order  to  express  the  relation 
between  the  respective  amounts  and  present  values  of  £1,  and 


THOMAN'S    TABLES  77 

of   £1    per    annum.       Thoman,    by    adopting    tlie    symbol    a", 
eliminates  the  factor  (R^  — 1)  altogether. 

The  factor  (R^  — 1)  may,  however,  be  ascertained  from 
Thoman's  tables  if  required  as  follows  : — - 

or  by  logarithms  : 

Log.  (EN -1)  =  Log.  RN  +  Log.  r-Log.a". 

and   Thoman's  a^  may  be   found   from   the   above   factors,    as 
follows  :  — 

Log.  o«  =  Log.  RN  +  Log.  7' -Log.  (R^-l). 

The  whole  of  the  formulae  previously  ascertained  by 
derivation  from  the  simple  formula  A  =  P  R^,  may  now  be 
expressed  in  terms  of  Thoman's  factors  of  R^,  and  a",  as 
follows  :  — 


Table  I.  RN        The  amount  of  £1  :  — 

will  be  expressed  by  Thoman's  factor      R^ 

Table  11.  zg-^      The  present  value  of  £1:  — 

will  be  expressed  by  Thoman's  factor     ^^ 

Table  111.     (  )     The  amount  of  £1  per  annum:  — 

It  has  been  proved  that :  — 
Table  Y     =  _™^ 
And  transposed,  it  will  be  seen  that :  — 

Ta^l^  III  =  TaHeT 
but  Table  I      =         RN 
and  Table  Y     =         a^ 
Therefore  Table  III  will  be  expressed 

by  Thoman's  symbols  -— — 


78  REPAYMENT    OF   LOCAL   AND    OTHER   LOANvS 

Table  IV.     (  —ry^ —  )  The  ijiesent  value  of  £1  per  annum  : - 

It  has  been  proved  in  Chapter  Till, 
that  Table  lY,  is  the  reciprocal 
of  Table  Y;  and  it  has  been 
shown  above,  that  Table  Y,  is 
expressed  by  Thomau's  symbol, 

Therefore  Table  lY  will  be  expressed 
by  Thonian's  factor 

Table  V.     ( -ns — ^  )     ^^'^  annuitij  irhich  £1  icill  purchase  :  — 

This,  as  shown  above,  is  the  equiva- 
lent of  Thoman's  factor  a"^ 

The  above  f  ormiilse  by  Thomau  may  be  stated  logarithmically 
as  follows,  using'  the  logs,  of  a"-  increased  by  10,  as  given  in 
Thoman's  tables:  — 

Table  T     =  R^  =Loo- R^' 

Table  //  =  J^  =  Log  1  =  0-  Log  R^' 

Table  111=  ^=  Log  R^  _  Log  a"  (add  10) 

Table  /F  =  —  =  Log  1  =  0-  Log  a"  (add  10) 
a^^ 

Table  V   =  a''  =  Log  a''    (deduct  10) 

and  any  problem  may  be  solved  by  adding  to,  or  deducting 
from,  the  log.  of  the  given  sum  or  annuity  the  logs,  of  E-^  and 
a"'  as  given  by  Thoman,  as  shown  in  the  various  standard 
calculation  forms  prepared  by  the  author  given  in  Chapter  X. 

In  using  the  above  log.  formulae  it  is  important  to  bear  in 
mind  the  previous  remarks  as  to  the  addition  of  10  to  the 
log.  of  a"'  in  the  tables  by  Thoman,  and  the  following  examples 
will  make  the  matter  clear.  It  affects  only  Tables  III,  lY 
and  Y. 

Taking  the  figures  relating  to  a  period  of  10  years  and  a  rate 
•of  accumulation  of  5  per  cent,  in  each  case,  the  above  log. 
formulae  will  now  be  applied  to  find  the  amounts  given  in 
those  tables,  which,  of  course,  relate  to  £1  only:  — 


THOMAN'vS    TABLES  79 

Table  HI.     Required  the  amount  of  £1  per  annum  for  10  rears 
at  5  per  cent. 

Calculation  (VI)  1. 

Log.  EN+10  ...     -10-211  S930 
deduct  Los:,  a'^    =   9112  2851 


1-099  6079 


which  is  the  Loff.  of £12-5779 


Table  IV.     Required  the  present  value  of  £1  per  annum  for 
10  years  at  5  per  cent. 

Calculation  (YII)  1. 

Log.  1  =  0+10...     =10000  0000 
dedtictLos.a^    =   9112  2851 


0-887  7149 


■which  is  the  Log.  of £7-7217 


I 


The  above  examples  show,  that  in  using  the  logs,  of  a"'  as 
given  in  Thoman's  tables  in  conjunction  with  the  log.  formulae 
relating  to  Tables  III  and  IT,  the  log.  of  fl"  (which  is  increased 
by  10  in  the  tables),  is  deducted,  and  consequently  10  must  be 
added  to  the  resulting  log.,  but  it  is  immaterial  where  10  is 
added.  In  the  above  example  10  has  been  added  to  the  log. 
of  H^  as  given  in  Thoman's  tables. 

In  the  case  of  Table  Y,  the  factor  a^  represents  the  values 
given  in  the  table,  and  as  Thoman's  logs,  are  increased  by  10, 
it  only  remains  to  deduct  10  therefrom  in  order  to  find  the  true 
log.  of  the  annuity  required. 

In  previous  chapters  dealing  with  Tables  I  to  Y  and  in  the 
following  chapters  dealing  with  other  calculations,  the  formulae 
and  rules  relating  to  the  method  by  Thoman  will  be  found  at 
the  head  of  each  chapter,  without  any  further  explanation. 


8o     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


CHAPTER   X. 

STANDARD  CALCULATION  FORMS,  PREPARED  BY 
THE  AUTHOR. 

The  five  published  tables  of  compound  interest.  The 
three  methods  of  calculation  in  each  form  and  the 
corresponding  rules  relating  thereto.  meaning  of 
all  symbols  and  factors,  and  the  various  methods  of 
finding  the  actual  log .  values.  six  standard  forms,  with 
references  to  the  above  and  list  of  problems  to  which 

EACH  MAY  BE  APPLIED.  AlSO  STANDARD  CALCULATION  FORMS 
TO  FIND  THE  EXACT  OR  APPROXIMATE  RATE  PER  CENT.  OR 
NUMBER  OF  YEARS,  IN  CONNECTION  WITH  EACH  OF  THE  FIVE 
TABLES  OF  COMPOUND  INTEREST  AND  THE  SINKING  FUND 
INSTALMENT. 


The  FIVE  STANDARD  TABLES  OF  COMPOUND  INTEREST  RELATING 
TO  THE  AMOUNT  AND  PRESENT  VALUE  OF  £1,  AND  OF  £1 
PER   ANNUM. 

Table      I.     The  amount  of  £1  in  any  number  of  years. 

Chapter  7T',  Form  7. 

Table    II.     The  present  value   of  £1  due  at   the   end   of  any 
number  of  years.  Chapter  F,  Form  II. 

Table  III.      The  amount  of  £1  per  annum  in   any  number  of 
years. 

Chapters  VI  and  XIII,  Forms  III  and  Ilia;. 

Table  IV.     The  present  value  of  £1  per  annum  for  any  number 
of  years.  Chapter  VII,  Form  IV. 

Table    V.     The    anmiity    which    £1    ivill    purchase    for    any 
number  of  years.  Chapter  VIII,  Form  V. 

In  these  tables  the  actual  values  only  are  generally  given, 
and  not  their  logarithmic  equivalents  which  are  found  in  the 
tables  of  M.  Fedor  Thoman. 


STANDARD    CALCULATION    FORMS  8i 

In  actual  practice,  extending  over  many  years,  the  autlior 
nas  repeatedly  felt  the  want  of  a  uniform  system  of  making  the 
various  calculations,  and  at  the  same  time  a  means  of  avoiding 
frequent  references  in  order  to  ascertain  the  particular  method 
to  be  adopted. 

The  absence  of  such  a  standard  was  not  felt  so  much  when 
the  problems  occurred  only  occasionally,  but  during  the  com- 
pilation of  this  book  such  a  large  number  of  calculations  had 
to  be  made  that  some  such  standard  became  imperative  if  only 
as  a  means  of  saving  the  labour  involved  in  re-writing  the  mere 
framework  of  each.  The  result  is  the  sis  standard  forms  used 
throughout  the  work  which  are  useful  not  only  as  blanks  to  be 
filled  up,  but  also  because  they  contain  the  formula  relating 
to  each  of  the  above  five  tables  of  compound  interest.  Each 
standard  form  includes  three  methods  of  calculation — namely 
(A)  by  formula,  (B)  by  the  published  mathematical  tables,  and 
(C)  by  Thoman's  tables,  and  these  methods  are  based  upon  the 
three  rules  stated  both  by  formula  and  in  words  at  the  head  of 
the  chapter  dealing  with  each  of  the  five  tables  of  compound 
interest.  In  order  to  bring  together  the  whole  of  the  formulae, 
symbols  and  rules,  as  well  as  the  methods  adopted,  a  series  of 
notes  have  been  made  showing,  firstly,  the  meaning  of  all  the 
symbols  and  factors,  and,  secondly,  the  method  of  finding  the 
various  factors  used  in  the  calculations.  The  object  of  doing 
this  is  to  render  this  chapter  a  complete  guide  to  anyone 
requiring  to  make  similar  calculations,  who  is  not  sufficiently 
interested  in  the  subject  to  become  more  fully  acquainted  with 
the  derivation  of  the  formulae.  Finally,  in  order  to  indicate 
the  particular  form  required  a  fairly  comprehensive  list  has 
been  prepared  of  the  problems  which  have  in  the  book  been 
solved  by  the  use  of  each  standard  form.  The  forms  are 
numbered  to  correspond  with  the  five  tables  of  compound  interest 
given  in  the  published  tables.  Each  form  contains  a  short 
heading  of  the  problem  as  stated  in  the  tables.  It  is  here 
advisable  to  mention  that  form  3x  for  finding  the  annual 
sinking  fund  instalment  is  based  upon  Table  III;  and  form  5 
may  be  used  to  find  the  equal  annual  instalment,  of  principal 
and  interest  combined,  to  repay  a  given  loan  in  a  stated  number 
of  years. 

If  any  of  the  calculations  involve  recurring  periods  less  than 
one  year,  the  necessary  alteration  may  be  made  by  taking  the 
number  of  periods  and  a  correspondingly  reduced  rate  per  cent., 
in  other  words,  the  rate  per  cent,  per  annum  in  the  published 
tables  becomes  the  rate  per  cent,  per  period.     This  is  where  the 


82     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

metliod  by  foriuula  Avill  be  found  invaluable,  inasmuch  as  in 
many  cases  the  rate  per  cent,  per  period  will  probably  not  be 
found  worked  out  in  any  of  the  published  tables. 

The  Three  Methods  of  Calculation  ix  Each  Form. 

A.  By  formula — Rule  1.     The  method  of  making  the  calcula- 

tions in  this  manner  is  fully  explained  in  the  following 
notes  and  in  the  standard  forms. 

B.  By  the  puhlished  tables  of  compound  Interest — Rule 2.  The 

first  step  is  to  ascertain  from  the  tables  the  actual  values 
relating  to  £1  under  similar  conditions  as  to  period  and 
rate  per  cent.  This  amount,  multiplied  by  or  divided 
into  the  sum  in  respect  of  which  the  calculation  is  to  be 
made,  gives  the  result  required. 

C.  By    Thoman's    Tables — Rule    3.       This    method    is    fully 

described  in  the  standard  forms,  and  consists  merely  of 
various  combinations  of  the  logs,  of  R^  and  <7"  and  of 
the  amounts  in  the  problem.  Thonian's  tables  are  fully 
described  in  Chapter  IX.  If  the  results  be  recjuired  to 
be  correct  to  the  utmost  decimal  point  the  method  by 
Thoman  should  be  adopted  because  these  tables  give  the 
actual  log.  values. 

A  full  explanation  of  the  rules  in  each  form  is  given  at  the 
head  of  the  chapter  dealing  with  the  subject  matter  of  each 
form. 

The  Rate  Per  Cent,  and  the  Number  of  Years.  In 
addition  to  the  six  standard  forms,  the  author  has  prepared  ten 
forms  showing  the  methods  of  determining  the  rate  per  cent, 
and  the  number  of  years  in  connection  with  each  of  the  five 
tables  of  compound  interest  and  the  sinking  fund  instalment. 
These  latter  forms  each  contain  particulars  of  an  example, 
worked  out  in  full  in  the  book  and  to  which  a  reference  is 
made. 

It  will  be  noticed  that  the  results  obtained  are  in  several 
■cases  approximate  only,  especially  as  regards  the  rate  per  cent. 
as  expressed  by  the  factors  R  and  r,  which,  however,  can  be 
determined  to  any  required  degree  of  accuracy  only  by  methods 
which  are  far  too  technical  to  be  included  in  a  work  of  this 
nature.  For  all  practical  purposes  an  approximation  of  the 
rate  per  cent,  is  sufficient,  and  this  may  generally  be  obtained 


STANDARD    CALCULATION    FORMS  83 

from  the  published  tables  of  conipoiuid  interest  giving  either 
the  actual  values  or  their  logarithmic  equivalents.  This 
difficulty  in  finding  the  exact  rate  per  cent,  arises  only  in  the 
case  of  an  annuity  or  other  periodic  payment.  In  the  case  of 
the  amount  and  present  value  of  a  sum  of  money  [Tables  I 
and  II]  the  calculation  is  a  simple  one  depending  upon  the  value 
of  11^,  If  the  value  of  this  be  known  as  well  as  one  of  the 
factors,  the  other  may  be  found  as  shown  in  the  following 
standard  forms  for  finding  the  rate  per  cent,  and  number  of 

/RN  — 1\ 

years.     The   factor  relating  to   all   annuities   is  (  ; —  j  and 

represents  the  amount  of  £1  per  annum  at  the  end  of  N  years. 
The  rate  per  cent,  cannot  be  determined  exactly  from  this 
factor,  and  the  method  by  approximation  is  too  long.  The 
practical  method  of  finding  the  approximate  rate  per  cent,  of 
accumulation  of  an  annuity  is  to  reduce  the  actual  example  to 
terms  of  an  annuity  of  £1  as  given  in  the  various  published 
tables  of  compound  interest.  Having  done  so,  a  reference  is 
made  to  the  tables  and  the  nearest  figure  ascertained.  This 
figure  is  adopted  in  the  calculation,  and  if  necessary  the  result- 
ing error  is  calculated  and  corrected.  An  example  of  this  may 
be  found  on  referring  to  Chapter  XXXII,  where  the  effect  of 
taking  the  equated  period  at  23,  instead  of  23' 136  years,  is 
fully  explained  and  accounted  for. 

Other  Methods  of  Calculatiox.  In  addition  to  the 
problems  which  may  be  solved  by  means  of  the  standard  forms 
included  in  this  chapter  there  are  several  others  which  occur  in 
the  course  of  the  book,  and  which  are  fully  explained  as  they 
arise.  In  all  cases  the  calculation  has  been  made  in  such  a 
form  that  it  may  be  adapted  to  any  similar  problem,  and  it  is 
not  necessary  to  repeat  the  forms  here.  The  folloAving  list 
includes  the  whole  of  these  special  calculations  :  — 

(1)  A  stated  sum  is  required  to  be  set  aside  and  accumulated 

at  compound  interest  for  a  stated  number  of  years.  At 
the  end  of  that  time  the  annual  sum  ceases,  but  the 
amount  then  in  the  fund  continues  to  accumulate  for  a 
further  stated  term.  The  rate  of  accumulation  may  be 
varied  or  not  at  the  end  of  the  first  period.  It  is  required 
to  ascertain  the  amount  in  the  fund  at  the  end  of  the 
second  period.  Statement  XVI.  D.  1. 

(2)  A  stated  amount  is  required  to  be  provided  at  the  end  of  a 

prescribed  period  by  the  accumulation  of  an  annual  sum 


B4     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

to  be  set  aside  during-  the  early  years  of  such  period  only. 
The  amount  then  in  the  fund  will  continue  to  accumulate 
until  the  end  of  the  prescribed  period.  It  is  required  to 
ascertain  the  annual  amount  to  be  set  aside  during  the 
first  part  of  the  prescribed  period. 

Calculation  (XXXIY)  G. 
(3)  A  stated  annual  sum  to  be  set  aside  for  a  prescribed 
period  and  accumulated  at  a  definite  rate  per  cent.,  is 
sufficient  to  provide  a  stated  amount  at  the  end  of  that 
period.  Before  the  expiration  of  the  period  a  change  is 
made  in  the  conditions  affecting  either  the  period  or  the 
rate  per  cent.,  or  both,  but  not  the  amount  to  be  provided. 
It  is  required  to  ascertain  by  one  calculation  the  equiva- 
lent future  annual  sum  under  the  amended  conditions. 

Statement  XXVI.  D. 

Symbols.  The  following  is  a  complete  list  of  the  symbols 
used  in  the  various  formulae,  and  in  the  standard  calculation 
forms,  with  the  meaning  of  each  symbol,  all  of  which  have  been 
fully  described  in  Chapter  III :  — 

A  denotes  the  amount  or  ultimate  sum  to  which  the  present 
sum  P  will  accumulate  in  X  years  at  the  ratio  or  common 
factor  R.  It  represents  both  the  ultimate  sum,  to  which 
a  stated  present  sum  will  amount  at  the  end  of  the  period, 
as  well  as  the  stated  sum,  due  at  the  end  of  the  period, 
of  which  the  present  value  P  is  required. 

P  denotes  the  principal  sum  in  hand ;  and  represents  also 
the  present  value  of  a  definite  sum  of  money  A  due  at  the 
end  of  a  stated  period  of  years ;  it  denotes  also  the  present 
value  of  an  annuity  or  other  periodic  sum,  Ay,  payable 
at  the  end  of  each  of  a  stated  number  of  years  or  periods 
X. 

R  denotes  the  ratio  or  common  factor  existing  between  each 
term  of  the  progression,  or  the  amounts  of  £1  at  the  end 
of  each  succeeding  year.  It  is  in  all  cases  £1  increased 
by  interest  upon  £1  for  one  year  at  the  rate  per  cent,  in 
question.  It  corresponds  with  the  algebraical  factor  r. 
r  denotes  the  interest  upon  £1  for  one  year  or  period  at  the 
stated  rate  per  cent.  It  is  always  less  than  the  above 
factor  E,  bv  unity. 

R-l  =  r. 

The  actual  rate  per  cent,  is  never  used  in  calculations 
involving  compound  interest,  but  is  always  expressed  in 


STANDARD    CALCULATION    FORMS  85 

its  relation  to  £1  ouly,  as  II,  and  /■,  above.  This  term  r 
is  not  tlie  equivalent  of  the  algebraical  factor  r,  in  a 
geometrical  progression. 

N  denotes  the  number  of  years,  or  other  equal  periods,  and 
must  not  be  confounded  with  the  factor  n  in  the 
algebraical  formula  for  a  geometrical  progression  which 
represents  the  number  of  terms  in  the  progression.  For 
this  reason  it  is  expressed  by  a  capital  letter.  This  term 
is  the  equivalent  of  the  algebraical  term  n  —  1. 

Ay  denotes  the  annuity  or  other  periodic  sum  to  be  paid,  set 
aside  or  received  at  the  end  of  each  year  or  period,  N. 

M  denotes  the  sum  to  which  the  annuity  or  other  periodic 
sum  Ay  will  amount,  if  accumulated  for  a  stated  number 
of  years  or  periods,  N,  at  a  stated  rate  per  cent. 

Formulae,  The  above  symbols  are  combined  in  various 
ways  in  the  formulse  given  in  the  book  resulting  in  various 
factors,  and  the  following  list  has  been  prepared  in  order  to 
show  the  meaning  of  such  factors,  and  also  the  manner  in 
which  they  may  be  found,  not  only  in  actual  values,  but  also  in 
their  logarithmic  equivalents. 

The  numbers  in  brackets  in  the  author's  standard  calculation 
forms  in  this  chapter  refer  to  the  following  notes  :  — 

Note  Symbol  or  factor  Remarks 

(1)  E,  =  ratio.  This  may  be  found  by  adding  to  £1, 

interest  upon  £1  for  one  year;  and 
the  log.  of  E-  may  be  found  from  the 
log.  tables  or  from  the  special  table 
of  those  logs,  given  in  Chapter  V. 

(2)  7-=  interest  upon     This  may  be  found  by  deducting  unity 

£1  for  one  year.         from  the  value  of  R  above,  and  the 
log.  of  r  may  be  found,  as  in  note  (1). 

(3)  RN.  This     symbol     corresponds     with     the 

symbol  r«  of  Thoman.  The  log.  of  R^ 
is  found  by  multiplying  the  log.  of  R 
by  the  number  of  years  or  periods. 
The  logs,  of  RN  are  given  in 
Thoman's  tables.  The  actual  values 
of  R^  are  given  in  Table  I. 


86  REPAYMENT    OF    LOCAL   AND    OTHER   LOANS 

Note  Symbol  or  factor  Kemaiks 

1  This  factor  Avill  very  rarely,  if  ever,  be 

(3a)  EN.  required  to  be  found  by  calculation. 

The  actual  values  are  g-iveu  in  Table 
II.  The  log.  of  this  factor  may  be 
found  by  deducting  the  log.  of  H^ 
from  0. 

(4)  IlN_i^  The  actual  values  of  this  factor  are  not 

given  in  Inwood  or  other  published 
tables,  although  they  may  be  found 
by  deducting  unity  from  the  values 
given  in  Table  I.  The  log.  of  this 
factor  is  not  given  in  Thoman's  tables, 
but  may  be  found  by  converting  the 
log.  of  EN  there  stated  into  its 
equivalent  ordinary  number  or  anti- 
log.,  deducting  unity  therefrom,  and 
finding  the  log.  of  the  remainder. 
The  log.  of  E^-l  so  found  may  be 
proved  by  Inwood,  by  deducting 
unity  from  the  amount  given  in 
Table  I,  and  finding  the  log.  of  the 
remainder.  The  log.  of  E^-l  may 
be  found  by  Thoman's  tables  as 
follows  :  — 

Log.  EN  +  log.  r+lO-log.  a", 
log.   7'  being  found,   as  explained  in 
note  (2)  above. 

(5)  RN_i,  tpjj^g    actual    values    of   this    factor    are 

r  given  in  Table  III  in  Inwood.     The 

logs,  of  this  factor  may  be  found  by 

Thoman's  tables  as  follows:  — 

Log.  EN +  10 -log.  a'K 

(6)  EN-1.  The    actual    values    of    this    factor    are 

EN  r.  given  in  Table  IV  in  Inwood  or  other 

similar  tables.  The  logs,  of  this  factor 
may  be  found  from  Thoman's  tables 
by  deducting  the  log.  of  iV\  there 
given,  from  10. 


STANDARD    CALCULATION    FORMS  87 

Note  Symbol  or  factor  Remarks 

R^  ^"  The    actual    values    of    this    factor    are 

(7)  li^  — 1.  given  in  Table  Y  in  Inwood  or  other 

similar  tables.  The  logs,  of  this 
factor  may  be  found  from  Thoman's 
tables  by  deducting  10  from  the  log. 
of  a"  there  given. 

(8)  <2".  This   is   a   term   employed   by   Thoman 

to  denote  the  annuity,  £a  per  annum, 
which  £1  will  purchase  for  n  years; 
and  the  actual  values  of  Avhich  are 
given  in  Table  Y.  The  logs,  given 
in  Thoman's  table  are  as  explained  in 
Chapter  IX,  the  true  logs,  of  a" 
increased  by  10.  The  relations  between 
a"  and  the  above  symbols  are  ex- 
plained briefly  in  the  foregoing  notes 
and  fully  in  Chapter  IX.  This  factor 
is  extremely  useful  for  finding  the 
equal  annual  instalment  of  prin- 
cipal and  interest  combined  (the 
annuity  method,  Chapter  XII). 


Standard  Calculation  Form,  No,  1. 

Table  I.     To  find  the  future  amount  of  a  present  sum. 

Chapter  lY. 
This  form  has  been  used  in  the  solution  of  problems  of  the 
following  nature  :  — 

Calculation. 

To  find  the  amount  of  loan  which  will  be  provided 
by  the  future  accumulation  of  the  present 
investments  representing  a  sinking  fund     ...  (^'^)  4. 

To  find  the  amount  of  loan  which  will  be  un- 
provided for  if  an  ascertained  deficiency  in  a 
sinking  fund  remains  uncorrected (XY)  6. 

To  find  the  amount  of  loan  which  will  be  provided 
by  the  future  accumulation  of  the  proceeds 
of  sale  of  assets  paid  into  the  fund       (XYII)  3. 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Standard  Calculation  Form,  No.  1, 

Table  I.     To  find  the  future  amount  of  a  present  sum. 
The  following  rules  are  explained  at  the  head  of  Chapter  lY. 
Here  state  the  general  nature  of  the  problem.  Calculation 

No. 

Here  state  full  details  of  the  actual  problem. 


(A)     By  Formuh 


A  =  P  RN 


Eulel. 


Log.  ^ 


Log.  Ratio  (1)   R 

MulHijly  Log.  R  by     N 


(3) .  RN 

Log.  Present  Sum  P 

fl^^Log.RNaboTe(3)    RN 


Y  allies . 


Required  future  amount,  £ 


Loqs. 


(B)     By  Table  I. 


A=  PRN 


Rule  2. 


Table  I.  years      per  cent. 

Amount  of  £1  (3) 

add  Loff.  Present  Sum 


RN 

P 


A 


Required  future  amount,  £ 


(C)     By  Thoman's  Table.  A  -  P  RN 

per  cent.  years 


Rule  3. 


Log.  Present  Sum 
add  Log.  RN  (3) 


P 

RN 


Required  future  amount,  £ 


STANDARD    CALCULATION    FORMS 


89 


The  Amount  and  Present  Value  of  One  Pound. 


Tables  I  and  II. 
To  find  the  number  of  years 


Standard  Forms,  1  and  2. 


based  on  Calculation  (XVI)  5. 


Given  factors : 

Present  sum 

Amount  thereof    .. 
Rate  per  cent. 

Ratio       

Interest  of  £1 

Details  of  Method : 

find          

find,  and  deduct  .. 

P 
A 

R 

r 

.     Log.  A 
.     Log.  P 

.     Log.  RN 

.     Log.  R. 

9463-00 
1123907 
31 
1035 
0-035 

11239-07 
9463-00 

1-035 

4-0507305 
3-9760288 

difference    . . 

00747017 

find          

0-0149403 

To  find  the  number  of  years,  divide  the  above  log.  of  R^  by 
the  above  log.  of  R,  as  described  in  Chapter  XXXII, 
and  the  quotient  is  the  number  of  years  required,  in 
this   case,    5   years. 

The  Amount  and  Present  Value  of  One  Pound. 
Tables  I  and  II.  Standard  Forms,  1  and  2. 

To  find  the  rate  per  cent : 


based  on  Calculation  (XV)  4. 


Given  factors : 

Present  sum P 

Amount  thereof    ...  A 

Number  of  years  ...  N 

Details  of  method  : 

find          Log.  A 

find,  and  deduct  ...  Log.   P 

difference    ...  Log.  R^ 

divide  this  log.   by 

the  number  of  years  N 

which  is  the  log.  of     ...  R 


946300 

14799-71 

13 


14799-71 
9463-00 


13 
1-035 


4-1702533 
3-9760288 

0-1942245 
0-0149403 


To  find  the  rate  per  cent.,  deduct  unity  from  the  above  ratio 
and  multiply  the  remainder  by  100,  or  ^  per  cent. 


90     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Standard  Calculation  Form,  No.  2, 

Table  II.     To  find  the  present  yalne  of  a  sum  due  at  a  future 
date.  Chapter  Y. 

This  form  has  been  used  in  the  solution  of  problems  of  the 
following  nature:  — 

Calculation. 

To  find  the  sum  now  payable  ^Yhich  is  the  equiva- 
lent of  a  given  loan  payable  at  the  end  of  a 
prescribed  number  of  years. 

vSee  Chapter  XXXII. 

Givea  a  stated  sum,  to  find  the  accumulated  amount 
of  an  annual  instalment,  to  be  set  aside  for  a 
limited  period  only;  the  amount  so  found  to 
accumulate  for  a  further  stated  period,  and 
then  amount  to  the  stated  sum.  "  The  method 
by  step"      (XVI)  3. 

The   annual   instalment    is  then   found   by 
means  of  standard  form,  No.  3x (XA  I)  4. 

The  methods  of  finding  the  rate  per  cent,  and  the  number 
of  years  are  similar  to  those  given  under  Table  I. 


STANDARD  CALCULATION  FORMS 


91 


Standard  Calculation  Form,  No.  2, 

Table  II.     To  find  the  present  value  of  a  sum  due  at  a  future 
date. 

The  following  rules  are  explained  at  the  head  of  Chapter  Y. 

Here  state  the  general  nature  of  the  problem.  Calculation 

No. 
Here  state  full  details  of  the  actual  problem. 


(A)     By  Formula.                          P  = 

A 
EN 

Eulel. 

/-Log.  Eatio                 (1) 
Log        MultiplijLog.'Rhj 

I                                  (3) 

E 

Values. 

Logs. 

EN 

Log.  Future  vSum 
deduct  Log.  E^  above 

(3) 

A 
EN 

P 

Eequired   present  v. 

ilue, 

£ 

• 

(B)     By  Table  II.                           P  = 

A 

EN 

Eule  2. 

Table  II.           years     per  cent. 

Present  value  of  £1  (3a) 

add  Log.  Future  Sum 

1 

EN 

A 

P 

Eequired   present  v 

alue, 

£ 

(C)     By  Thoman's  Table.              P  = 
percent.            years 

A 
EN 

Eule  3. 

Log.  Future  Sum 
deduct  Log.          (3) 

A 
EN 

P 

Eequired  present  value,    £ 


92     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Standard  Calculation  Form,  No.  3. 


Table  III.     To  find  tlie  amount  of  an  annuity.  Chapter  VI. 

This  form  has  been  used  in  the  solution  of  problems  of  the 
following  nature  :  — ■ 

Calculation. 

To  find  the  amount  which  should  stand  to  the 
credit  of  a  sinking  fund  at  any  time  during 
the   repayment  period      (■^^^)  2. 

To  find  the  amount  of  loan  which  will  be  provided 
by  the  future  accumulation  of  :  — 

(1)  the     original     annual     sinking     fund 

instalment       (XY)  5. 

(2)  the  additional,  augmented,  or  reduced 

annual  sinking  fund  instalment (^^^I)  2. 

(3)  the    income    from    the    present    invest- 

ments representing  the  fund       (XIX)  1. 

(4)  the  annual  increment  of  the  fund     ...        (XIX)  4. 


STANDARD    CALCULATION    FORMvS 


9S 


Standard  Calculation  Form,  No,  3. 
Table  III.     To  find  the  amount  of  an  annuity. 
The  following  rules  are  explained  at  the  head  of  Chapter  YI. 
Here  state  the  general  nature  of  the  problem.  Calculation 


Here  state  full  details  of  the  actual  problem. 


Required  future  amount,   £ 
(B)     By  Table  III.  M  =  A^y  (^^~^^ 

Table  III.  years     percent.     W^-l 


Amount  of  £1  per  annum  (5) 
add  Log.  Annuity 


^y 


M 


Required  future  amount,  £ 


No. 


(A)     By 

Fornnila. 

M  =  A,  (1^^- 

^) 

Rulel. 

Log.  Ratio 
Multiply  Log. 

Convert  Log. 
to  ordinary  n 
deduct  unity 

Log.  of  this  is 

(1) 
Rby 

(3) 
umber 

(4) 

above 

(4) 

■       (2) 

R 

Values. 

Logs. 

Loo- 

RN 

W—l^ 

RN 

-1 

1- 

RN-1 

Log.  Annuity 
af/fZLog.RN_] 

deduct  Log.  7 

A.y 

RN-1 

Ay(RN- 
r 

-1) 

M 

Rule  2. 


(C)     By  Thoman's  Table.      M  =  Ky  ("^L^) 
per  cent.  years 


Rules. 


Log.  Annuity 
add  Log.  RN  in 

Table +10     (3) 


deduct  Log .  a^      (8) 


Ay 
RN 


AyRN 

a^ 


M 


Required  future  amount,  £ 


94 


REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 


The  Amount  of  One  Pound  per  Annum. 

Table  III.  Standard  Form  3. 

To  find  the  number  of  years  : 

based  on  Calculation  (XYIII)  4. 


Given  factors  : 

Annuity          

Amount  of  annuity 
Rate  per  cent. 

Ratio       ...      

Interest  of  £1 
Details  of  Method : 

find           

find,  and  deduct  .. 

R 

r 

Log.    M 
Log.   Ay 

Log.  r 
Log.  RN- 

RN 

Log.  RN 
Log.  R 

7500-00 

57468-48 

300 

1-03 

0-03 

57468-48 
750000 

-1 

0-22987 
1- 

4-7594297 
3-8750613 

difference    . . 
find,  and  add 

0-8843684 
2-4771213 

the  sum  is     .. 

i-3614897 

find  value  of  tliis  log 
add  unity 

whicb  is  tlie  value  of  .. 
find          

1-22987 

0-0898606 

find          

00128372 

To  find  the  number  of  years,  divide  the  above  log.  of  RN  by 
the  above  log.  of  R,  as  described  in  Chapter  XXXII, 
and  the  quotient  is  the  number  of  years  required,  in 
this  case,  7  vears. 


The  Amount  of  One  Pound  per  Annum 

Table  III. 
To  find  the  rate  per  cent : 


Standard  Form  3. 
based  on  Calculation  (XVIII)  7. 


Given  factors ; 

Amount  of  annuity 

M 

1176-58 

Annuity          

A^/ 

58-3715 

Number  of  years  . . . 

16 

Details  of  method : 

find           

Log.  M 

1176-58 

30706241 

find,  and  deduct  ... 

Log.  A// 

58-3715 

1-7662008 

diiference    . . . 

1-3044233 

find  value  of  this  log. 

20- 1569 

which  is  the  amount  of  an  annuity  of  one  pound  for  16  years 
at  the  required  rate  per  cent. 


vSTANDARD    CALCULATION    FORMS  95 

To  ascertain  tlie  rate  per  cent.,  refer  to  Table  III,  giving 
the  amounts  of  one  pound  per  annum,  and  find  the  nearest 
value  to  the  above  amount  of  20' 1569  in  16  years.  If  the 
rate  so  found  is  not  near  enough,  refer  to  Thoman's  tables 
and  find  the  nearest  log.  to  1-3044233,  which  is  ascertained 
by  deducting  the  log.  of  o"  from  the  log.  of  E^,  plus  10. 

Eequired  rate  per  annum,  3  per  cent. 

Note.  In  cases  where  the  rate  per  cent,  is  not  included  in 
the  published  tables  of  compound  interest,  or  in  Thoman's 
tables,  the  above  method  will  give  only  approximate  results. 


Standard  Calculation  Form,  No.  3x. 

Table  III.     To  find  the  annual  sinking  fund  instalment. 

Chapter  XIII. 
This  form  has  been  used  in  the  solution  of  problems  of  the 
following  nature  :  — 

Calculation. 

To  find  the  annual  sinking  fund  instalment,  to  be 
set  aside  out  of  revenue  or  rate,  and  accumu- 
lated at  compound  interest,  to  repay  a  stated 
loan  at  the  end  of  a  prescribed  period (^^)  1- 

To  find  the  annuity  which  Avill  amount  to  a  stated 
sum  in  any  number  of  years 

To  find  the  additional  annual  sinking  fund 
instalment  required  to  provide  the  amount  of 
loan  which  will  be  unprovided  for  owing  to  a 
deficiency  in  the  amount  in  the  fund (^^^I)  1- 

To  find  the  amount  by  which  the  original  annual 
sinking  fund  instalment  may  be  reduced  in 
consequence  of  the  withdrawal,  during  the 
repayment  period,  of  part  of  the  loan  from 
the  operation  of  the  fund (XA  III)  1. 

To  find  the  future  annual  increment  to  be  added 
to  the  fund,  and  accumiilated  at  compound 
interest,  to  provide  the  balance  of  loan,  which 
will  not  be  provided  by  the  future  accumula- 
tion of  the  present  investments  representing 
the  fund      (XYI)9. 


96 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Standard  Calculation  Form,  No.  3x, 
Table  III.     To  find  the  animal  sinking  fund  instalment. 
The  following  rules  are  explained  at  the  head  of  Chapter  XIII. 

Here  state  the  general  nature  of  the  problem.        .       Calculation 

No. 
Here  state  full  details  of  the  actual  problem, 

(A)     By  Formula  Ai/ =  M    (jp^)  Rulel. 


Log   . 
W—1 

'  Log.  Eatio                 (1) 
Multi'ply  Log.  E  by 

(3) 
Convert  Log. 

to  ordinary  number 
deduct  unity 

Log.  of  this  is        (4) 

E 

N 

EN 

Values. 

Logs. 

EN 

-1 

EN-1 

Log.  Amount  of  Loan 
add  Log.  r             (2) 

M 

r 

deduct  Log.  (E^  _  \\  above 

(4) 

Mr 
EN-1 

Ay 

Eequired  annual  instalment,  £ 

M 

{^) 

By  Table  III.              Ai/  =     Kn_i 

r 

Eule  2. 

Log.  Amount  of  Loan 
Table  III.  years     percent. 

Amount  of  £1  per  annum  (5) 
deduct  Log. 


M 
EN-1 


Av 


Eequired  annual  instalment,  £ 


(C)     By  Thoman's  Table.      Ay  ^  M   ^^\ 
per  cent.  years 


Eule  3. 


Log  Amount  of  Loan 
add  Log.  a«  (8) 


deduct  Log.  EN  in 
Table +  10       (3) 


.AI 


Ma" 
EN 


Ay 


Required  annual  instalment,  £ 


STANDARD    CALCULATION    FORMvS 


97 


The  Sinking  Fund  Instalment. 
Table  III.  Standard  Form,  3x. 


To  find  the  number  of  years 


based  on  Calculation  (XV)  1. 


Given  factors : 

Amount  of  loan    ...  M 

Annual    instalment  Aiy 
Rate  per  cent. 

Ratio       ^ 

Interest  of  £1       ...  ^ 


26495 

680-234 
31 

1035 
0-035 


Details  of  Method 


find          

find,  and  deduct  ... 

Log.  M. 
Log.  Ay 

Log.  /■ 
Log.  RN-1 

RN 

.     Log.  RN 
.     Log.  R 

26495 
680-234 

1-36324 
1- 

4-4231639 
2-8326581 

difference   . . 
find,  and  add 

1-5905058 
2-5440680 

the  sum  is     .. 

find  value  of  tbis  log 
add  unity 

0-1345728 

vvbicb  is  tbe  value  of  .. 
find          ...      ...      .. 

2-36324 

0-3735087 

find          

00149403 

To  find  tbe  number  of  years,  divide  tbe  above  log.  of  R^  by 
tbe  above  log.  of  R,  as  described  in  Cbapter  XXXII, 
and  tbe  quotient  is  tbe  number  of  years  required,  in 
tbis  case,  25  years. 


gS     REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


The  Sinking  Fund  Instalment. 


Table  III. 
To  find  the  rate  per  cent 

Given  factors : 


Standard  Form,  3x. 
based  on  Calculation  (XY)  1. 


Amount  of  loan    ... 
Annual     instalment 
Number  of  years  ... 

Details  of  method  ; 

find          

find,  and  deduct  ... 

L 
L 

Ay 
N 

lo;.  M. 
Og-.  Ay 

26495 
680-234 
25 

26495 
680-234 

38-94986 

4-4231639 
2-8326581 

difference    ... 
find  value  of  this  log. 

1-5905058 

wliieli  is  tlie  amount  of  loan  which  will  be  provided  by  an 
annual  instalment  of  one  pound  for  25  years  at  the 
required  rate  per  cent. 

To  ascertain  the  rate  per  cent.,  refer  to  Table  III,  giving 
the  amounts  of  one  pound  per  annum,  and  find  the  nearest 
value  to  the  above  amount  of  38-94986  in  25  years.  If 
the  rate  so  found  is  not  near  enough,  refer  to  Thoman's 
tables  and  find  the  nearest  log.  to  1-5905058  which  is 
ascertained  by  deducting  the  log.  of  (7»  from  the  log.  of  E^, 
plus  10. 

Eequired  rate  per  annum,  3|  per  cent. 

Note.  In  cases  where  the  rate  per  cent,  is  not  included  in 
the  published  tables  of  compound  interest,  or  in  Thoman's 
tables,  the  above  method  will  give  only  approximate  results. 

Standard  Calculation  Form,  No.  4. 

Tablk  IY.     To  find  the  present  value  of  an  annuity. 

Chapter  YII. 

This  form  has  been  used  in  the  solution  of  problems  of  the 
following  nature:  — 

Calculation. 

To  find  the  sum  now  payalde  which  is  the  equiva- 
lent of  tlie  future  annual  sinking  fund 
instalments  to  be  set  aside  to  repay  a  given 
loan  at  the  end  of  a  prescribed  period  of 
years;  and  for  which  such  annual  instalments 
might  be  redeemed        Sec  Chapter  XXXII. 


STANDARD    CALCULATION    FORMS 


99 


Standard  Calculation  Form,  No.  4. 

Table  IV.     To  find  the  present  value  of  an  annuity. 

The  following  rules  are  explained  at  the  head  of  Chapter  VII. 

Here  state  the  general  nature  of  the  problem.  Calculation 


Here  state  full  details  of  the  actual  problem. 


(A)     By  Formula. 


^y  {-w^) 


Required   present  value,    £ 


No. 


Rule  1. 


^Log.  Ratio                 (1) 
Multiply  Log.  R  by 

(3) 
Convert  Log. 

to  ordinary  number 
deduct  Unity 

Log.  of  this  is        (4) 

Values. 
R 

N 

Logs. 

Log    . 
R^— 1 

RN 

RN 

-1 

RN-1 

Log.  Annuity 

add  Log.  (RN-1) 
above          (4) 
deduct  Log.  R^  above 

(3) 

Ay 
RN_1 

RN 

deduct  Log.  r       (2) 

r 

P 

(B)     By  Table  IV 


/R^^-l\ 


Table  IV.         years      per  cent. 
Present  Value  £1  per 

annum  (6) 

add  Log.  Annuity 


RN-1 


RNr 

Ay 


Required   present   value,    £ 


(C)     By  Thoman's  Table. 

per  cent.  years 


a" 


Required  present  value,   £ 


Rule  2. 


Rule  3. 


Log.  Annuity 

add  10 
deduct  Log.  a" 

(«) 

Ay 

an 

p 

REPAYMENT    OF   LOCAL   AND    OTHER   I<OANS 


The  Present  Value  of  One  Pound  per  Annum. 
Table  IV.  Standard  Form.  4. 


To  find  the  number  of  years  : 


based  on  Calculation  (XVIII)  14. 


Given  factors : 

Annuity          

Ky 

40-215 

Present  value  thereof 

P 

313118 

Rate  per  cent. 

3 

Eatio      

R 

103 

Interest  of  £1 

r 

003 

Details  of  method : 

find         

Log.P 

313-118 

2-4957086 

find,  and  deduct  ... 

Log.  Ky 

40-215 

1-6043881 

difference    . . . 

0-8913205 

find  value  of  this  log. 

7-7861 

which   is  the  present  value  of   an   annuity  of   one   pound,    at 
3  per  cent.,  for  the  required  number  of  years. 


To  ascertain  the  number  of  years,  refer  to  Table  IV  giving 
the  present  values  of  one  pound  per  annum,  under  3  per 
cent.,  and  find  the  nearest  value  to  the  above  amount  of 
7-7861.  If  the  rate  per  cent,  is  not  given  in  the  tables, 
refer  to  Thoman's  tables,  under  the  nearest  rate  per  cent., 
and  find  the  nearest  log.  to  0-8913205,  which  is  found  by 
deducting  the  log.  of  a"  there  given  from  10. 

Required  period,  9  years. 

Note.  In  cases  where  the  rate  per  cent,  is  not  included  in 
tbe  published  tables  of  compound  interest,  or  in  Thoman's 
tables,  the  above  method  will  give  only  approximate  results. 


STANDARD    CALCULATION    FORMS 


OP  THE 

UNIVERSJTV 

OF 


The  Present  Value  of  One  Pound  per  Annum. 


Table  IV. 


Standard  Form.  4. 


To  find  the  rate  per  cent : 

based  on  Calculation  in  Statement  XXXII.  D. 


Given  factors : 

Annuity          

A 

1725-58 

Present  value  thereof 

P 

28374-73 

Number  of  years  ... 

N 

23 

Details  of  method : 

find          

Log.P 

28374-73 

4-4529318 

find,  and  deduct  ... 

Log.  Ay 

1725-58 

3-2369352 

difference    . . . 

1-2159966 

find  value  of  this  log. 

16-4436 

which  is  the  present  value  of  an  annuity  of  one  pound  for 
23  years,  at  the  required  rate  per  cent. 

To  ascertain  the  rate  per  cent,  refer  to  Table  IV,  giving  the 
present  values  of  one  pound  per  annum,  and  find  the 
nearest  value  to  the  above  present  value  of  16-4436  in 
23  years.  If  the  rate  so  found  is  not  near  enough,  refer 
to  Thoman's  tables  and  find  the  nearest  log.  to  1-2159966, 
which  is  ascertained  by  deducting  the  log.  of  a"  there 
given  from  10. 

Eequired  rate  per  annum,  3  per  cent. 


Note.  In  cases  where  the  rate  per  cent,  is  not  included  in 
the  published  tables  of  compound  interest,  or  in  Thoman's 
tables,  the  above  method  will  give  only  approximate  results. 


102  REPAYMENT   OF    LOCAL   AND    OTHER   LOANvS 


Standard  Calculation  Form,  No.  5. 

Table  V.  To  find  the  annuity  whicli  a  present  sum  will 
puicLiase,  or  the  annuity  of  which  £1  is  the 
present  value.  Chapter  YIII. 

This  form  has  been  used  in  the  solution  of  problems  of  the 
following  nature  :  — 

Calculation, 

To  find  the  equal  annual  instalment  of  principal 
and  interest  combined,  to  be  paid  to  the 
lender,  in  order  to  repay  a  stated  loan  in  a 
prescribed  period        (XII)  4. 

To  find  the  amount  by  which  the  original  annual 
sinking  fund  instalment  may  be  reduced  in 
consequence  of  :  — 

(1)  a    surplus    in    the    fund    owing    to    an 
excessive  past  accumulation  of  the  fund  (XYIII)  10. 

(2)  a  surplus  in  the  fund,  due  to  the  pay- 

ment into  the  fund  of  any  sum  not 
provided  out  of  revenue  or  rate, 
namely  :  — 

(a)  the  proceeds  of  sale  of  part  of 
the  assets  representing  the 
security  for  the  loan (XVII)  1. 

or 

(h)  a  realised  profit  upon  the  sale 
of  an  investment  representing 
the  fund. 

To  find  the  additional  sinking  fund  instalment,  to 
be  set  aside,  and  added  to  the  fund  during 
tlie  unexpired  portion  of  tlie  repayment 
period,  to  compensate  for  a  deficiency  in  the 
amount  now  in  the  fund (^^  )  ^' 


STANDARD    CALCULATION    FORMS 


103 


Table  V.  To  find  tlie  annuity  which  a  present  sum  will 
purchase,  or  of  which  it  is  the  present  value. 
To  find  the  equal  annual  instalment  of  principal 
and  interest  combined. 

The  following  rules  are  explained  at  the  head  of  Chapter  VIII. 

Here  state  the  general  nature  of  the  problem.  Calculation 

No. 

Here  state  full  details  of  the  actual  problem. 


(A)     By  Formula.  At/ 


/W  r  \ 


Values. 


Log 


Log.  Ratio  (1) 

Multiply  Log.  R  by 

Convert  Log. 

to  ordinary  number 
deduct  unity 


(4) 

Log.  Present  Sum 
add  Log,  RN  above  (3) 
Log.  r  (2) 

deduct  Log.  (R^  - 1)  above 

(4) 


RN 


RN 

-1 


RN-1 


p 

RN 

r 


RN-1 


Ay 
Required  annuity,  £ 


Table  V.  years      percent. 

Annuity  £1  will  purchase  (7) 
add  Log.  Present  Sum 


RN 

RN  - 
P 


Ay 


Required  annuity,  £ 


Rule  1. 


Logi 


(B)     By  Table  Y.  ^V  ^  ^^    (^ZTl)  Rule  2. 


(C)     By  Thoman's  Table.  Ai/  =  P  an 

per  cent.  years 


Required  annuity,  £ 


Rule  3. 


Log.  Present  Sum 
add  Log.  a^            (8) 

P 

deduct  10 

A?/ 

I04    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


The  Annuity  which  One  Pound  will  Purchase 

and 

The  Equal  Annual  Instalment  of  Principal  and  Interest 

Combined. 

Table  V.  Standard  Form,  5. 


To  find  the  number  of  years  : 


based  upon  Calculation  (XV)  3. 


Given  factors : 

^ 

Present  sum 

P 

469-74 

Annuity          

Ky 

45-594 

Rate  per  cent. 

31 

Ratio       

R 

1-035 

Interest  of  £1 

r 

0-035 

Details  of  method : 

find         

Log.  Ay 

45-594 

1-6589086 

find,  and  deduct  ... 

Log.P 

469-74 

2-6718612 

difference    . . . 

2'9870474 

find  value  of  tbis  log. 

0-097061 

which  is  the  annuity  which  one  pound  will  purchase  at  3^  per 
cent,  for  the  required  number  of  years. 

To  ascertain  the  number  of  years  refer  to  Table  V,  giving  tbe 
annuity  which  one  pound  will  purchase,  under  3^  per 
cent.,  and  find  the  nearest  value  to  the  above  amount  of 
0- 097061.  If  the  rate  per  cent,  is  not  given  in  the  tables, 
refer  to  Thoman's  tables,  under  the  nearest  rate  per  cent., 
and  find  the  nearest  log.  of  a"  to  2-9870474.  This  may 
be  ascertained  by  an  inspection  of  the  mantissa  only. 


Required  period,  13  years. 

Note.  In  cases  where  the  rate  per  cent,  is  not  included  in 
the  published  tables  of  compound  interest,  or  in  Thoman's 
tables,  the  above  method  will  give  only  approximate  results. 


STANDARD    CALCULATION    FORMS 


105 


The  Annuity  which  One  Pound  will  Purchase 

and 

The  Equal  Annual  Instalment  of  Principal  and  Interest 

Combined. 
Table  V.  Standard  Form,  5. 


To  find  the  rate  per  cent : 


based  upon  Calculation  (XVIII)  10. 


Given  factors  : 

Present  sum 

Annuity          

Number  of  years  . . . 

P 

Ay 

N 

447-27 
57-4446 
9 

Details  of  method  : 

find          

find,  and  deduct  . . . 

L( 
L( 

Dg.   Ay 

57-4446 
447-27 

0-128434 

1-7592493 
2-6505698 

difference    . . . 
find  value  of  tbis  log. 

T-1086795 

wbicb  is  tbe  annuity  wbicb  one  pound  will  purcbase  for  9  years 
at  tbe  required  rate  per  cent. 

To  ascertain  tbe  rate  per  cent.,  refer  to  Table  V,  giving  tbe 
annuity  wbicb  one  pound  will  purcbase,  and  find  tbe 
nearest  value  to  tbe  above  annuity  of  0' 128434  in  9  years. 
If  tbe  rate  so  found  is  not  near  enougb,  refer  to  Tboman's 
tables  and  find  tbe  nearest  log.  of  a"  to  1-1086795.  Tbis 
may  be  ascertained  by  an  inspection  of  tbe  mantissa  only. 

Required  rate  per  annum,  3  per  cent. 

Note.  In  cases  wbere  tbe  rate  per  cent,  is  not  included  in 
tbe  published  tables  of  compound  interest,  or  in  Tboman's 
tables,  tbe  above  metbod  will  give  only  approximate  results. 


Section  II. 


The  Methods  of  Repayment  of  the  Loan  Debt 

of  Local  Authorities  and  Commercial 

and  Financial  Undertakings. 


log 


CHAPTER  XI. 

THE  REPAYMENT  OF  THE  LOAN  DEBT  OF  LOCAL 
AUTHORITIES  AND  COMMERCIAL  AND  FINAN- 
CIAL UNDERTAKINGS. 

Alternative  methods  allowed  by  the  Public  Health  Act, 
1875,  AND  other  Acts.  Comparison  of  methods  as 
regards  the  actual  repayment  to  the  lender,  and  the 
annual  charge  against  Revenue  or  Rate. 

The   Instalment  Method. 

BY  AN  EQUAL  ANNUAL  INSTALMENT  OF  PRINCIPAL  TO  BE  REPAID 
TO  THE  LENDER.  No  SiNKING  FuND  REQUIRED,  BUT  AN 
EQUAL  PERIODICAL  REPAYMENT  OF  PRINCIPAL.  AnNUAL 
CHARGE     AGAINST     THE     REVENUE     OR     RaTE     AcCOUNT     OF 

successive  years  composed  of  an  equal  amount  of 
principal  and  a  gradually  decreasing  amount  of  interest. 
Statement  showing  the  final  repayment  of  the  Loan. 


Having  obtained  a  series  of  rules  and  forinulse  relating  to 
all  problems  involving  compound  interest,  they  will  now  be 
applied  to  problems  of  actual  finance,  beginning  witb  the  re- 
payment of  the  loan  debt  of  local  authorities.  This  affords  a 
very  good  subject  for  treatment  by  the  mathematical  method, 
not  only  on  account  of  the  variety  of  the  problems  occurring  in 
actual  practice,  but  because  the  original  conditions  and  regula- 
tions are  of  a  fairly  uniform  character.  This  uniformity  is 
much  more  pronounced  than  is  the  case  with  the  loan  debt  of 
commercial  and  financial  undertakings,  where  not  only  much 
more  variable  and  elastic  conditions  exist,  but  also  greater 
facilities  to  alter  the  original  arrangements  between  the  borrower 
and  the  lender. 

Excluding  for  the  present  the  provisions  contained  in  early 
Acts  of  Parliament,  which  vary  considerably,  the  general 
principles  now  in  force,  and  governing  the  matter,  are  contained 
in  Section  234  of  the  Public  Health  Act  of  1875.  These 
provisions  may  be  accepted  as  the  standard  now  adopted  in  all 


no    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

public  general  Acts,  special  Acts,  and  provisional  orders  of  the 
Local  Government  Board.  There  are  of  course  small  variations 
in  detail,  but  the  principle  remains  in  all  cases  substantially  the 
same,  and,  with  the  exception  of  the  introduction  of  a  new  form 
of  "  non-accumulating  "  sinking  fund,  there  has  not  been  any 
change  since  1875.  These  provisions  relate  to  the  repayment 
of  the  debt,  and  will  apply  equally  to  commercial  and  financial 
undertakings,  as  they  are  based  upon  general  financial  practice. 
There  is  not  anything  new  in  the  methods  laid  down  in  the  Act, 
but  the  merit  of  the  section  lies  in  the  fact  that  for  the  first 
time  definite  methods  were  prescribed  in  place  of  the  very 
varied  practice  previously  followed.  To  state  an  extreme  case, 
borrowing  powers  will  never  be  granted  in  future  without  any 
obligation  whatever  as  to  redemption.  This  section  specifies 
three  alternative  methods,  at  the  option  of  the  local  authority, 
by  which  the  loan  debt  may  be  repaid,  and  provides  in  effect 
that :  — 

(a)  The  local  authoritv  shall  repay  the  moneys  so  borrowed 
by:~  ■ 

(1)  equal  annual  instalments  of  principal,  or  by 

(2)  equal    annual    instalments    of    principal    and 

interest  combined ; 

or  (h)  The  local  authority  shall  in  every  year  set  apart  as  a 
sinking  fund  and  accumulate  in  the  way  of  compound 
interest  such  a  sum  as  will,  with  accumulations  in  the 
way  of  compound  interest,  be  sufficient  to  pay  off  the 
moneys  so  borrowed  within  the  period  sanctioned. 

The  alternative  methods  of  repayment  are  usually  described 
as  :  — 

[a)   (1)   The    instalment   method. 
(2)  The  annuity  method. 

(h)   The  sinking  fund  method. 

The  sinking  fund  method  in  the  Act  of  1875  is  the  same 
as  the  accumulating  sinking  fund  referred  to  in  the  1893 
clauses  of  the  Local  Government  Board. 

The  sections  of  the  Public  Health  Act,  1875,  and  other  Acts 
relating  to  the  borrowing  of  money  by  local  authorities  lay 
down  five  distinct  principles,  namely:  — 

The  power  to  borrow  is  limited  to  works  of  a  more  or  less 
permanent  nature. 


THE    INSTALMENT    METHOD  iii 

Tlie  amount  to  be  borrowed  is  limited. 

The  period  of  repayment  must  be  fixed  by  tbe  Local  Govern- 
ment Board,  having  regard  to  the  relative  permanency  of 
the  works. 

The  Public  General  Acts  contain  a  provision  that  the  period 
sanctioned  by  the  Local  Government  Board  for  the 
repayment  of  any  loan  shall  not  in  any  case  exceed  a 
period  prescribed  in  each  Act. 

The  amount  annually  required  to  discharge  the  liability  in 
respect  of  interest  upon  the  loan  and  the  repayment  of 
the  debt  is  chargeable  against  the  rate  or  revenue  account 
of  each  year. 

There  are  two  main  distinctions  to  be  drawn  between  the 
above  methods  (1)  as  regards  the  actual  repayment  of  the  loan, 
and  (2)  as  regards  the  charge  upon  the  rate  or  revenue  accounts 
of  the  successive  years  of  the  repayment  period.  The  instalment 
and  annuity  methods  both  provide  for  the  actual  repayment  to 
the  lender  each  year  of  a  definite  proportion  of  the  loan  or  of 
the  loan  and  interest  combined.  The  sinking  fund  method,  on 
the  other  hand,  contemplates  the  provision  annually  of  an 
instalment  of  such  amount  as  will,  if  set  aside,  invested,  and 
accumulated  for  the  prescribed  period,  provide  for  the  repay- 
ment of  the  loan  in  one  sum  at  the  end  of  the  period.  Power  is 
given,  however,  under  certain  conditions  to  apply  part  of  the 
sinking  fund  in  repayment  of  the  loan  during  the  prescribed 
period   of  repayment. 

Each  of  these  methods  will  be  considered  in  detail,  taking 
first :  — 

The  Instalment  Method,  in  which  the  loan  is  repaid  to  the 
lender  by  equal  annual  instalments  of  principal  only,  and 
interest  is  paid  to  him  upon  the  balance  of  loan  unpaid.  This 
me-thod  applies  mainly  to  advances  made  to  local  authorities  by 
the  Public  Works  Loan  Commissioners,  and  also  to  loans  by 
the  larger  insurance  companies  to  the  Metropolitan  boroughs 
and    other   local   authorities. 

This  method  is  also  commonly  used  by  commercial  and 
financial  undertakings,  and  is  known  as  the  deferred  payment 
system.  The  hire  purchase  system,  on  the  other  hand,  is  a 
commercial  form  of  the  annuity  method.  The  instalment 
method  is  exceedingly  simple  in  operation,  seeing  that  it  is 
merely  an  arithmetical  calculation,  and  does  not  involve  any 
question  of  compound  interest  whatever. 


112         REPAYMENT   OF   LOCAL   AND   OTHER   LOANvS 

Generally  the  repayment  period  is  30  years  or  longer,  but 
in  order  to  simplify  the  problem  and  to  enable  a  comparison  to 
.be  made  with  the  annuity  and  sinking  fund  methods  to  be 
hereafter  described,  in  all  cases  the  example  will  relate  to  the 
repayment  of  a  loan  of  £1,000,  in  ten  years,  with  interest  at 
5  per  cent.,  which  rate  will  be  assumed  to  be  payable  to  the 
lender,  and  will  also  be  the  rate  of  accumulation  of  the  sinking 
fund. 

As  will  be  seen  by  the  following  statement,  the  municipality 
will  repay  to  the  lender  at  the  end  of  the  first  year  :  — 

^/ 10  of  the  principal £100 

Interest  on  £1,000 50 

£150 

and  this  amount  will  be  charged  to  the  rate  or  revenue  account. 

At  the  end  of  the  second  year  the  municipality  will  repay  to 
the  lender :  — 

^ 1 10  of  the  principal,  as  before       ...      £100 
Interest  on  £900       45 

£145 

and  so  on  each  year  until,  at  the  end  of  the  tenth  year,  they  will 

repay  :  — 

^ 1 10  of  the  principal,  as  before       ...      £100 
Interest  on  £100      5 

£105 

the  effect  being,  as  regards  the  municipality,  that  the  rate  or 
revenue  account  will  be  charged  year  by  year  with  a  gradually 
decreasing  amount,  to  the  relief  of  the  later  generations  of 
ratepayers. 

As  regards  the  lender,  he  originally  advanced  to  the 
municipality  a  sum  of  £1,000,  which  is  repaid  to  him  at  the 
rate  of  £100  per  annum,  which  will  require  to  be  invested  each 
year,  with  the  result  that  his  income  will  be  constantly 
fluctuating,  and  he  will  hold  a  number  of  small  investments 
instead  of  one  large  one.  The  following  statement  (XI.  A.), 
shows  the  operation  of  the  repayment  from  year  to  year  by 
means  of  the  constant  instalment  of  £100  of  principal,  and  also 
shows  the  decreasing  amount  of  interest  annually  paid  to  the 
lender.  As  the  principal  and  the  interest  are  both  charged  to 
the  revenue  or  rate  account,  it  shows  the  annually  decreasing 
loan  charge. 

Similar  statements  will  be  given  relating  to  the  annuity  and 
sinking  fund  methods;  and,  finally,  a  statomont  will  be  pn^pared 


THE    INSTALMENT    METHOD 


113 


comparing  tlie  effect  of  the  repayment  by  all  three  methods. 
In  the  case  of  the  instalment  and  annuity  methods  there  is  not 
any  accnmulating'  sinking  fund,  and  therefore  there  is  not  any 
complication  arising  from  the  rate  of  accnmnlation.  This  will 
be  dealt  with  under  the  head  of  the  sinking  fund  method. 
The  instalment  method,  so  far  as  regards  the  actual  repajanent 
to  the  lender,  is  exactly  similar  to  the  ordinary  repayment  of 
debt  by  commercial  and  finaneial  undertakings,  and  there  is  not 
any  difference  in  principle  if  the  instalments  are  not  equal  in 
amount  or  are  made  at  unequal  intervals  of  time.  The  lender 
receives  interest  each  year,  upon  the  actual  balance  owing  to 
him,   since  the  last  date  to  which  interest  has  been  paid. 

Unlike  the  annuity  and  instalment  methods,  there  is  not 
any  variation  in  the  calculation  if  the  interest  be  paid  half- 
yearly  or  otherwise  instead  of  yearly.  It  is  a  simple  arith- 
metical calculation  not  complicated  in  any  degree  by  compound 
interest. 

STATEMENT  XI.  A. 

The  Repayment  of  Debt  of  Local  Authorities.  The  Instalment 
Method. 

Showing  the  repayment  of  a  Loan  of  £1,000  in  10  years,  with 
interest  at  5  per  cent,  by  equal  annual  instalments  of 
principal  only. 


Owing  at 


begin- 
ning 
of 
year. 

Interest 
at 

5% 

Total 

owing. 

Repayments. 

Balance 
owing 
at  end 
of  year. 

Charge  to  Ei 

EVENUE. 

ear 
nd. 

Princi- 
pal.   ] 

[nterest 

Total. 

Princi- 
pal. 

Interes 

.    Total. 

Yea 

1 

1000 

50 

1050 

100 

50 

150 

900 

100 

50 

150 

1 

9 

900 

45 

945 

100 

45 

145 

800 

100 

45 

145 

2 

8 

800 

40 

840 

100 

40 

140 

700 

100 

40 

140 

3 

4 

700 

35 

735 

100 

35 

135 

600 

100 

35 

135 

4 

5 

600 

30 

630 

100 

30 

130 

500 

100 

30 

130 

5 

6 

500 

25 

525 

100 

25 

125 

400 

100 

25 

125 

6 

7 

400 

20 

420 

100 

20 

120 

300 

100 

20 

120 

7 

8 

300 

15 

315 

100 

15 

115 

200 

100 

15 

115 

8 

9 

200 

10 

210 

100 

10 

110 

100 

100 

10 

110 

9 

0 

100 

5 

105 

100 

5 

105 

— 

100 

5 

105 

10 

1000 

275 

— 

1000 

275 

1275 

— 

1000 

275 

1275 

114         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


CHAPTER  XII. 

THE  REPAYMENT  OF  THE  LOAN  DEBT  OF  LOCAL 
AUTHORITIES  AND  COMMERCIAL  AND  FINAN- 
CIAL UNDERTAKINGS  {Coiitinued). 

The  Annuity  Method, 

by    an    equal    annual    instalment    of    principal    and 
interest  combined,  to  be  repaid  to  the  lender. 

Methods  of  calculating  the  annual  instalment  by  formula 
AND  tables;  and  the  general  rules  based  thereon.  No 
sinking  fund  required,  but  an  equal  periodical  repay- 
ment TO  the  lender  of  principal  and  interest  combined. 
The  relation  between  such  equal  annual  instalment, 
the  sinking  fund  instalment  (Chapter  XIII)  and  the 
equal  annual  instalment  of  principal  only  (Chapter  XI). 
Statement  showing  the  final  repayment  of  the  Loan. 

Author's  Standard  Calculation  Form,  No.  5. 


Formulae. 

The  whole  of  the  formuhe  at  the  head  of  Chapter  VIII, 
relating  to  Table  V  {the  annuity  which  £1  will  pwrhase)  apply 
to  the  present  method. 

General    Rules    deduced    from    the    formulae     relating    to 
Table  V  :— 

To  find  the  equal  annual  instalment  of  principal  and  interest 
combined,  to  repay  a  given  loan  during  a  stated  period. 

Author's  Standard  Calculation  Form,   No.  5. 

Rule  1.     If  the  rate  per  cent,  he  not  given  in  Table  V,  or  in 
Thoman's  Tables  :  — 

Proceed  by  the  foriinda  relating  to  Table  Y . 

Calculation  {XII)  4  A. 

Rule  2.     If  the  rate  per  cent,  be  given  in  Table  F  :  — 

Multiply  the  annuity  given  in  the  table  by  the 
amount  of  the  loan.  The  product  is  the  required 
annual  instalment.  Calculation  {XII)  4  B. 


THE    ANNUITY    METHOD  115 

Rule  3.     If  the  rate  yet  cent,  he  given  in  Thoman's  Table  :  — 
To  the  log.  of  the  amount  of  the  loan  add  the  log. 
of  a",  as  given  by  Thoman :  deduct  10  from  the  sum. 
of   the   logs.        The   remainder   is    the    log.    of   the 
required  annual  instalment.      Calculation  {XII)  4  C . 

Rule  4.  Find  the  sinking  fund  instalment  by  any  of  the  rules 
given  in  the  following  chapter;  add  to  the 
instalment  so  found  one  year's  interest  upon  the 
loan.  The  rate  pier  cent,  in  both  cases  to  be  the  rate 
of  interest  to  be  paid  to  the  lender.  The  sum  is  the 
required  annual  instalment.  Calculation  {XII)  5. 

To  find  the  rate  per  cent,  or  number  of  years ^  proceed  as 
shown  in  the  standard  form  for  the  purpose,  relating  to  Table  \  , 
given  m  Chapter  X. 


The  Annuity  Method.  Under  this  method  there  is,  as  in 
the  case  of  the  instalment  method,  an  actual  repayment  each 
year  to  the  lender,  the  whole  of  which  is  charged  to  the  rate  or 
revenue  account  of  each  year.  But  in  this  case  the  lender 
receives  an  equal  amount  each  year,  composed  of  principal  and 
interest  combined.  To  the  extent  that  it  involves  an  equal 
annual  charge  upon  the  rate  or  revenue  account  of  the  muni- 
cipality during  the  whole  of  the  repayment  period,  it  is  an 
improvement  upon  the  instalment  method,  but  it  still  has  not 
any  advantage  to  the  lender.  As  will  be  seen  by  the  detailed 
statement,  XII.  A.,  following,  and  also  by  the  comparative 
statement  in  Chapter  XIII,  after  the  sinking  fund  method,  the 
annual  amount  repaid  to  the  lender  consists  of  an  increasing 
amount  of  principal  and  a  decreasing  amount  of  interest ;  and, 
further,  if  the  lender  be  a  trustee,  or  requires  for  any  purpose  to 
allocate  the  repayment  as  between  capital  and  income,  he  must 
make  a  somewhat  difficult  calculation.  The  lender  has  to  re- 
invest each  year  a  gradually  increasing  amount  of  principal, 
unless  he  sets  aside  an  equal  annual  proportion  of  the  amount 
paid  to  him,  as  a  sinking  fund,  as  will  be  explained  later. 

The  formulae  and  tables  will  now  be  applied  to  ascertain  the 
equal  annual  instalment  of  principal  and  interest  combined 
required  to  repay  a  loan  of  £1,000  in  10  years  at  5  per  cent. 
There  are  several  ways  of  doing  this,  but  the  clearest,  although 
not  the  shortest,  will  be  first  described,  being  the  one  which 
best  illustrates  the  principles  involved.  Leaving  the  actual 
calculation  for  the  moment,  the  transaction  will  be  divided  into 


ii6    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

two  parts,  ignoring  for  the  present  the  annual  repayments  to 
the  lender.  The  loan  of  £1,000,  if  not  repaid,  M-ill  accumulate 
at  5  per  cent,  compound  interest  with  yearly  breaks,  and  at 
the  end  of  the  period  will  amount  to  £1628'9,  as  shown  by 
Calculation  (XII)  1.  The  next  step  is  to  ascertain  the  amount, 
at  the  end  of  10  years  at  5  per  cent.,  of  an  equal  annual  instal- 
ment of  £1  per  annum,  or  12'5779,  as  shown  by  Calculation 
(XII)  2. 

It  has  now  been  ascertained  that  £1,000  in  10  years  at 
5  per  cent,  will  amount  to  £1628-9  and  that  each  £1  of  equal 
annual  instalment,  or  annuity,  will  at  the  end  of  10  years 
amount  to  £12"57T9 ;  and  it  is  therefore  obvious  that  the  equal 
annual  instalment  required  will  be,  in  sterling  or  other 
currency,  exactly  the  number  of  times  that  £125779  is  con- 
tained in  £1628-9.  By  dividing  £16289  by  £12-5779,  the 
required  equal  annual  instalment  of  principal  and  interest 
combined  is  obtained,  viz.,  £129-51,  as  shown  by  Calculation 
(XII)  3.  The  actual  details  of  the  above  Calculations  (XII)  1 
and  (XII)  2  are  given  at  the  end  of  the  chapter  upon  the 
author's  standard  calculation  forms,  No.  1  and  No.  3,  both  of 
which  are  made  by  three  methods  :  — 

A.  by  formula. 

B.  by  the  published  tables; 
and  C.  by  Thoman's  tables. 

The  two  factors  referred  to  have  now  been  ascertained:  — 
Calculation  (XII)  1  shows  that  the  original  loan  of  £1.000  will 
in  10  years,  at  5  per  cent.,  amount  to  £1628-90:  and  Calcula- 
tion (XII)  2  shows  that  £1  per  annum  will  in  10  years,  at 
5  per  cent.,  amount  to  £12-5779;  and  the  required  annual 
instalment  of  principal  and  interest  combined  is  obtained  by 
dividing    £162890    by    £12-5779    by    logarithms    as    follows. 


CALCULATION  (XII)  3. 

To  find  the  equal  annual  instalment  of  principal  and  interest 
combined  to  repay  a  given  loan. 

Required  the  equal  annual  instalment  of  principal  and 
interest  combined,  to  be  repaid  to  the  lender  as  and  when 
set  aside,  to  repay  £1,000  in  10  years  at  5  per  cent. 

By  Tables  I  and  III  and  Logs.  Based  on  CUilculations  (XII)  1 
and  (XII)  2. 


THE    ANNUITY    METHOD 


117 


Table  I,  Calculation  (XII)  1 : 

Amount   of    £1,000    in    10    years    at 

5  per  cent 

deduct,  log. 
Table  in,  Calculation  (XII)  2  : 

Amount  of  £1  per  annum  for  10  years 
at  5  per  cent 


1628-9 


3-2118930 


12-5779     10996079 


2-1122851 


which  is  the  log.  of  the  required  equal  annual  instal- 
ment of  principal  and  interest  combined,  viz.,        £129-51 

The  principle  involved  in  the  above  method  of  ascertaining 
the  equal  annual  instalment  of  principal  and  interest  combined, 
to  be  repaid  to  the  lender  as  shown  in  Calculation  (XII)  3,  is, 
that,  on  the  one  hand,  there  is  the  original  loan  of  £1,000 
quietly  rolling  up  all  by  itself  for  the  prescribed  period ;  and, 
on  the  other  hand,  there  is  an  equal  annual  instalment  of 
£12951  also  rolling  up  at  the  same  rate  per  cent,  for  the  same 
period.  The  annual  instalment  is  of  such  amount  that,  at  the 
end  of  the  period  both  accounts  will  amount  to  exactly  the  same 
sum.  Seeing  that  the  rate  of  accumulation  is  the  same  in  both 
cases,  it  is  obvious  that  the  transfer  from  one  account  to  the 
other  of  an  annual  sum  in  repayment  of  principal  and  interest 
combined,  out  of  the  accumulating  credit,  may  be  made  without 
in  any  way  affecting  the  result  arrived  at  by  considering  the 
two  factors  as  independent  transactions. 

The  following  table  shows  the  methods  of  finding  the  equal 
annual  instalment  of  principal  and  interest  combined  in  the 
foregoing  example,  and  also  demonstrates  the  derivation  of  the 
formula  relating  to  Table  Y  from  the  formulae  relating  to 
Tables  I  and  III :  — 


Numerator  :  — 

The  amount  of  £1  in  any 
number  of  years    

Denominator  :  — 

The  amount  of  £1  per  annum, 
in  the  same  number  of 
years          


By  Published 

Tables. 


Actual 

Values. 


Table  I     1628-90 


Table  III      125779 


By 

Formulae. 


RN 


KN- 


Table  V  = 


Table  I 
Table  m 


129-51 


RN    T 


ii8    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

In  Chapter  IX  it  was  pointed  out  tliat  the  above  formula 
^j^ — y    is  the  equivalent  of  Thoman's  factor  (a"),  both  of  which 

denote  the  annuity  which  £1  will  purchase  for  any  number  of 
years.  In  Chapter  VIII  the  same  formula  was  arrived  at,  by 
deduction,  from  Table  IV,  which  gives  the  present  values  of  an 
annuity  of  £1.  The  present  example  proves  that  the  formula 
for  Table  \^  may  be  also  found  by  deduction  from  Tables  I  and 
III,  and  consequently  that  the  equal  annual  instalment  of 
principal  and  interest  combined  may  be  found  by  Table  V  or  by 
Thoman's  factor  («")  in  a  much  more  direct  way  than  by  using 
Tables  I  and  III  as  above.  The  calculation  will  therefore  be 
made  by  Table  V,  on  the  author's  standard  calculation  form 
No.  5,  using  the  three  methods  therein  contained,  namely,  by 
formula,  by  the  published  table,  and  by  Thoman's  method. 
See  Calculation  (XII)  4  at  the  end  of  this  chapter. 

The  general  rules  relating  to  each  method  are  given  at  the 
head  of  this  chapter. 

The  Relation  between  the  Equal  Annual  Instalment  of 
Principal  and  Interest  Combined,  and  the  Sinking  Fund 
Instalment.  On  comparing  the  above  Calculation  (XII)  13, 
relating  to  the  annuity  method,  with  Calculation  (XIII)  1,  in 
the  following  chapter  by  which  the  sinking  fund  instalment  is 
ascertained,  it  will  be  seen  that  Calculation  (XIII)  1  is  the 
simpler  because  it  involves  only  one  reference  to  the  published 
tables  (No.  III.).  The  present  comparison  is  made  in  order 
to  compare  the  annual  instalment  of  principal  and  interest 
combined  with  the  sinking  fund  instalment,  ignoring  the  fact 
that  there  is  a  more  direct  method  of  finding  the  equal  annual 
instalment  of  principal  and  interest  combined  by  means  of 
Table  V.  It  will  be  seen  that  the  equal  annual  instalment  of 
[)rincipal  and  interest  is  greater  than  the  sinking  fund  instal- 
ment by  £50,  which  is  one  year's  interest  upon  the  loan  of 
£1,000  at  5  per  cent,  per  annum. 

The  equal  annual  instalment  of  principal  and  interest 
combined,  to  be  paid  to  the  lender  under  the  annuity  method 
may  bo  therefore  ascertained  in  the  following  manner:  — 

First  ascertain  the  sinhing  fund  instalment  ivhich  will 
provide  the  loan  at  the  end  of  the  period,  as  in 
Calculation  [XIIT)  1 ,  taldng  as  the  rate  of  arc^lVlulation 
of  the  sinlnng  fund  the  rate  of  interest  to  he  paid'  to  the 
lender  under  the  annuity  method. 


THE    ANNUITY    METHOD  119 

Then  add  to  the  annual  sinkiny  fund  instalment  so  found  one 
year's  interest  upon  the  loan  at  the  same  rate,  which  is 
the  rate  of  interest  payable  to  the  lender^  and  the  result 
is  the  equMl  annual  instalment  of  principal  and  interest 
combined  under  the  annuity  method,  as  follows  :  — 

CALCULATION  (XII)  5. 

To  find  the  equal  annual  instalment  of  principal  and  interest 

combined  to  repay  a  given  loan. 
Required    the    equal    annual    instalment     of     principal     and 

interest  combined,  to  be  repaid  to  the  lender  as  and  when 

set  aside,   to  repay  £1,000  in  10  years  with   interest  at 

5  per  cent. 
Based  on  the  Sinking  Fund  Method.     See  Calculation  (XIII)  1. 

Amount   of   the    annual   instalment   by    the  sinking 

fund  method  as  found  by  Calculation  (XIII)  1,     £  79-51 
Add  one  year's  interest  on  £1000  at  5  per  cent.  6000 

£129-51 

which  is  the  required  annual  instalment  of  principal 
and  interest  combined,  as  found  by  Calculations 
(XII)  3  and  (XII)  4. 

It  is  important  to  bear  in  mind  the  requirements  as  to  the 
rate  per  cent.  This  method  is  of  practical  use  only  in  finding 
the  equal  annual  instalment  of  principal  and  interest  combined 
under  the  annuity  method.  The  sinking  fund  instalment  is 
more  easily  found  by  the  direct  method  shown  in  Calculation 
(XIII)  1,  than  by  employing  the  method  shown  in  Calculation 
(XII)  3,  to  find  the  annual  instalment  of  principal  and  interest 
combined,  and  deducting  therefrom  the  amount  of  interest  upon 
the  loan. 

The  difference  between  the  two  instalments  may  be  stated  in 
terms  of  the  respective  formulae  by  deducting  the  formula 
relating  to  the  sinking  fund  instalment  from  the  formula 
expressing  the  equal  annual  instalment  of  principal  and  interest 
as  follows  :  — 


I20    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

wliicli  is  tlie  interest  upon  £1  for  one  year.  One  instalment 
may  also  be  expressed  in  terms  of  the  other.  If  it  be  required 
to  find  the  equal  annual  instalment  of  principal  and  interest, 
having  found  the  sinking  fund  instalment,  divide  the  formula 
relating  to  the  equal  annual  instalment  of  principal  and  interest 
by  the  formula  relating  to  the  sinking  fund  instalment  as 
follows  :  — 

thereby  proving  that  the  equal  annual  instalment,  of  principal 
and  interest  combined,  may  be  found  by  multiplying  the 
ascertained  sinking  fund  instalment  by  the  value  of  R^  as  given 
in' the  published  table  (No.  1);  or  by  logs.,  by  adding  to  the 
log.  of  the  sinking  fund  instalment  the  log.  of  E^,  as  given  in 
Thoman's  tables.  The  sum  of  the  above  logs,  is  the  log.  of  the 
equal  annual  instalment  of  principal  and  interest  combined. 

Applying  this  rule  to  the  present  example  :  — 

Log.  Sinking  fund  instalment =  795046       I'QOO  3921 

add  Log.  EN  10  years,  5  per  cent.       l" 62889     0-211  8930 

Log.  equal  annual  instalment  of  principal 

and  interest  combined        =  2' 112  2851 


which,  as  shown  by  Calculation  (XII)  4,  is  £12951. 

This  method  Avill  rarely  be  used  in  practice,  but  is  interest- 
ing as  furnishing  a  further  example  of  the  relation  between  the 
above  formulae. 

The  Eepayment  of  the  Loan  by  the  Annuity  Method. 
The  following  Statement  XII.  A.  shows  the  repayment  of  the 
loan,  year  by  year;  and  should  be  compared  Avith  the  similar 
Statement  XI.  A.,  relating  to  the  instalment  method  in  the 
previous  chapter.  It  should  also  be  compared  with  Statement 
XIII.  A.,  relating  to  the  sinking  fund  method  in  the  next 
chapter,  when  it  will  be  noticed  that  not  only  is  the  total 
annual  charge  to  the  revenue  or  rate  account  uniform  during 
the  whole  of  the  repayment  period  under  both  the  annuity  and 
sinking  fund  methods,  but  that  the  total  annual  charge  is  also 
the  same  in  amount  in  each  case  provided  that  the  rate  of 
accumulation  of  the  sinking  fund  is  the  same  as  the  rate  of 
interest  payable'  upon  the  loan.  The  following  Statement 
XII.  A.   also  shows  the  increasing  amounts  of  principal   and 


THE    ANNUITY    METHOD  121 

tile  couvsequent  decreasing-  amounts  of  interest  contained  in  the 
equal  annual  instalment  repaid  to  the  lender. 

After  considering  the  sinking  fund  method  in  the  following 
chapter  the  results  under  the  three  methods  will  be  shown  in 
tabular  form,  both  as  regards  the  actual  repayment  to  the 
lender,  and  also  the  annual  charges  to  the  revenue  or  rate 
account  during  the  successive  years  of  the  repayment  period,  in 
Statement  XIII.  B. 

The  two  methods  already  discussed,  namely,  the  instalment 
method  and  the  annuity  method,  involve  the  provision  each 
year,  out  of  rate  or  revenue  of  part  of  the  principal  and  interest, 
and  such  annual  provision  is  actually  repaid  to  the  lender  as 
and  when  set  aside.  They  do  not  tlierefore  in  any  way  partake 
of  the  nature  of  a  sinking  fund,  which  relates  only  to  the 
provision  for  the  repayment  of  the  principal  in  one  amount  at 
the  end  of  a  definite  period,  and  will  be  described  in  the 
following  chapter. 


REPAYMENT    OF    LOCAL   AND    OTHER   LOANS 


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THE    ANNUITY    METHOD 


123 


Calculation  (XII)  1. 

Standard  Calculation  Form,  A'o.  1. 

To  find  the  future  amount  of  a  present  sum. 

To  find  the  amount  which  will  be  owing  at  the  end  of  a  stated 
period  in  respect  of  a  given  loan  if  it  be  allowed  to 
accumulate  at  compound  interest.  Table  I. 

Required  the  amount  of  £1,000  at  the  end  of  10  years  at  5  per 
cent,  per  annum,  compound  interest. 


[A)     By  Formula, 


A=PRN 


Rule  1,  Chapter  lY. 


Log  r  Log.  Ratio 

R^   ^      Mxilti'ply  Log.  R  by 


Log.  Present  Sum 
add  Loff.  R^  above 


R 

N 

105 
10 

00211893 
10 

RN 

(1-05)10 

0-2118930 

P 

RN 

1000 

3- 
0-2118930 

A 

3-2118930 

Required  future  amount,  £1628-90. 


(B)     By  Table  I. 


A=PRN 


Rule  2,  Chapter  IV. 


Table  I.     10  vears,  5  per  cent. 

Amount  of  £1  j  R^ 

add  Los;.  Present  Sum 


A 


1-628895   0-2118930 
1000    3- 


3-2118930 


Required  future  amount,   £1628-90. 


(C)     By  Thoman's  Table.      A  =  P  RN 
5  per  cent.  10  years. 


Rule  3,  Chapter  lY 


Log.  Present  Sum 
aiZ^Log.RN 


P 
RN 


A 


1000 


3- 
0-2118930 


3-2118930 


Required  future  amount,  £1628-90. 


124    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Calculation  (XII)  2. 

Standard  Calculation  Foriii,  No.  3. 

To  find  the  amount  of  an  aunuity  in  any  number  of  years. 

Table  III. 
Required  the  amount  of  £1  yier  annum  for  10  years  at  5  per 
cent,  per  annum,  compound  interest. 

^RN ]^^^ 

(A)     By  Formula.  M  =  Ay  ( j        Rule  1,  Chapter  VI. 


^Log.  Ratio 

Multiply  Log.  R  by 

Convert  Log. 

to  ordinary  numboi' 
deduct  unity 

Log.  of  this  is 

Log.  Annuity 
add  Log.  RN  -  1  above 

deduct  Log.  r 

11 

N 

105 
10 

00211893 
10 

Log 
RN— 1 

RN 

(ro5)io 

0-2118930 

RN 

-1 

1-6289 
1- 

^ 

RN-1 

0-6289 

T-7985779 

A^ 

RN-1 

1- 

0- 

1^7985779 

A2/(RN- 
r 

-1) 

1-7985779 
2-6989700 

M 

1-0996079 

Required  amount,  £125779. 


(H)     liy  Table  III.       M  =  Ay  f^^^^      Rule  2,  Chapter  VI. 


Table  111.     10  years,  5  per  cent. 
Amount  of  £1  per  annum 
Add  Log.  Annuity 


RN-l     12-5779 


Ay 


M 


Required  amount,  £12-5779.    This  amount  is  given  in  Table  III. 


(C j     By  Thoman's  Table .     M  =  Ay  ^  ^'^^  j      Rule  3,  Chapter  VI . 
5  per  cent.  10  years. 


Log.  Annuity 
add  Log.  RN  iu 

Table +10 


deduct  Log.  a" 


Ay 

RN 


0- 
10-2118930 


Ai/RN 


10-2118930 
9-1122851 


M 


1-0996079 


Required  amount,  £12'5779. 


THE    ANNUITY    METHOD  125 

Calculation  (XII)  4. 

Standard  Calculation  Form,  No.  5. 

To  find  the  annuity  wliicli  a  present  sum  will  purchase  for  any 

number  of  years. 
To  find  the  equal  annual  instalment  of  principal  and  interest 

combined  to  repay  a  given  loan.     The  Annuity  Method. 

Table  Y. 
Required  the  equal  annual  instalment  of  principal  and  interest 

combined  to  l)e  repaid  the  lender  as  and  when  set  aside,  to 

repay  £1,000  with  interest  in  10  years  at  5  per  cent. 


A)     By  Formula.         %  =  Pf^^J       Rule  1,  Chapter  VIII 


Log.  Ratio 

Multiply  Log.  R  by 

Convert  Log. 

to  ordinary  number 
deduct  unity 

1{ 

N 

1-05 
10 

0-0211893 
10 

Log    . 
RN_1 

RN 

(1-05)10 

0-2118930 

RN 

-1 

1-62889 
1- 

RN- 

1 

0-62889 

T-7985779 

Log.  Present  Sum 
add  Log.  R^  above 
Log.  r 

p 

RN 

r 

1000 
0-05 

3- 
0-2118930 

2-6989700 

deduct  Log.  (RN  - 1)  above 

RN  - 

1 

1-9108630 
1-7985779 

Aw 

21122851 

Required  annuity,   £129-5046. 


(B)     By  Table  V.          Ay  =  P  (^^^1^  ^    Rule  2,  Chapter  YIII. 


Table  V.     10  years,  5  per  cent. 

Annuity  £1  will  purchase 

add  Log-.  Present  Sum 


RN 


RN-1 

p 


0  1295 
1000 


1- 11 22851 
3- 


Ay 


21122851 


Required  annuity,   £129-5046. 


(C)     By  Thoman's  Table.       At/^  P  a"         Rule  3,  Chapter  YIII. 
5  per  cent.  10  years. 

Log.  Present  Sum 
add  Loo".  a'^ 


deduct  10 


9-1122851 


121122851 


Ay 


21122851 


Required  annuity,  £129-5046. 


126         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


CHAPTER  XIII. 

THE  REPAYMENT  OF  THE  LOAN  DEBT  OF  LOCAL 
AUTHORITIES  AND  COMMERCIAL  AND  FINAN- 
CIAL UNDERTAKINGS  {Continued). 

The  Sinking   Fund   Method. 

BY  SETTING  ASIDE  AND  ACCUMULATING  AN  EQUAL  ANNUAL 
INSTALMENT  IN  OEDER  TO  PEOVIDE  THE  PRINCIPAL  ONLY  AT 
THE  END  OF  THE  REDEMPTION  PERIOD. 

I,   The  Accumulating  Sinking  Fund. 
Methods    oe    calculating    the    annual    instalment    by 

FORMULA  and  TABLES   AND   THE   GENERAL   RULES   BASED 

THEREON.  Description  of  the  method  and  the  calcula- 
tion  OF   the   annual   instalment.     Statement    showing 

THE  final  repayment  OF  THE  LOAN.  COMPARISON  OF  THE 
instalment,  ANNUITY  AND  SINKING  FUND  METHODS,  ILLUS- 
trated by  a  statement  showing  in  each  case  the  annual 
charge  to  revenue  or  rate. 

Author's  Standard  Calculation  Form,  No.  3x. 
2.   The  Non-accumulating  Sinking  Fund. 

The  OBJECT  of  the  fund  and  its  relation  to  the  METHODS 
PRESCRIBED  IN   SeC.   234  OF  THE  PuBLIC  HeALTH  AcT,   1875. 

Statement  showing  the  final  repayment  of  the  loan 
and  the  annual  charges  to  revenue  or  rate. 

Note. — Unless  it  is  otherwise  expressly  stated,  the  term 

"  sinking   FUND,"  WILL,  THROUGHOUT  THE  BOOK,    APPLY 
ONLY  TO   AN  ACCUMULATING    SINKING    FUND. 


Formulae. 

Variation  of  Table  III,     The  aiuiuifij  which  uill  amount  to 

£1  in  any  number  of  years,  or  ^   hTT^J'T 


THE    SINKING    FUND    METHOD  127 

A.  To  find   the   annuity   which    will   amount   to   £1   in   any 

TMimber  of  years  :  — 

(7)  Formula,  Ay=  (  t^n  _x  j 

by   logs.:      Log.     (^required    annuity)  =  Log.     r  — 
Log.  (EN-1) 

(2)  By  Thoman's  Method:  — 

Formula,  Ay=^^^ 

by  logs.:      Log.  (^reqtiired  annuity)  — Log,  a"  — 
{Log.  EN +10) 

B.  To  find  the  annual  sinking  fund  instalment  which  tvill 

amount  to  any  given  loan,  in  any  number  of  years  :  — 

(7)  Formula,  A.y  =  ^i(  ^        J 

by  logs.  :      Log.  (required  instalvient)=Log.  of  Loan  + 

Log.  r-Log.  (R^-1) 

(2)  By  Thoman's  Method:  — 

/ft™  \ 
Formula,  Ay  =  M  f  t^^  1 

by  logs.:      Log.   (required  instalment)  =  Log.   of 
Loan  +  Log.  a^-{Log.  RN+IQ) 

General  Rules  deduced  from  the  above  formulae. 

To  find  the  animal  instalment  to  be  set  aside  and  accumulated 
as  a  sinking  fund  to  repay  a  given  loan  at  the  end  of  a  prescribed 
7iumber  of  years.      Author's  Standard  Calculation  Form,  No.  3x. 

Rule  1.     If  the  rate  per  cent,   be  not  given  in  Table  III,  or 
in  ThoTnan's  Tables  :  — 

Proceed  by  the  formula  derived  from  Table  III,  as 
shoivn  above.  Calculation  {XI II)  1  A . 

fiule  2.     If  the  rate  per  cent,  be  given  in  Table  III :  — 

Divide  the  amount  of  the  loan  by  the  amount  given 
in  the  table.  The  qiiotient  is  the  required  annual 
instalment.  Calculation  {XIII)  1  B. 


128    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Rule  3.     If  the  rate  per  cent,  he  given  in  Th Oman's  Table  :  — 
To  the  log.  of  the  loan,  add  the  log.  of  a"  as  given  by 
Thoman.     Deduct  therefroiit  the  log.  of  W^  as  given 
by  Thoman;  also  deduct  10.      The  remainder  is  the 
log .  of  the  reqtiired  instalvient. 

Calculation  {XIII)  IC. 

To  fnd  the  rate  yer  cent,  or  number  of  years ^   proceed  as 
shown  in  the  standard  form  for  the  purpose,  given  inChapter  X. 


The  Accumulating  Sinking  Fund.  The  sinking  fund 
metliod  provides  for  the  setting  aside  each  year,  and  accumulat- 
ing by  way  of  compound  interest,  such  a  sum  as  will  be  sufficient 
to  pay  off  the  money  borrowed  within  the  prescribed  period.  It 
will  be  gathered  from  tbe  above  provision  (which  is  laid  down 
in  the  Public  Health  Act,  1875,  and  is  contained  in  principle 
in  all  subsequent  Acts)  that  this  method  differs  from  the  instal- 
ment and  annuity  methods  in  two  particulars,  viz.  :  — 

1.  It  provides  for  the  repayment  of  principal  only,   and  is 

quite  apart  from  any  question  of  interest  on  the  loan. 

2.  The  repayment  of  the  principal  money  is  not  made  by 

instalments,  but  takes  place  at  tlie  end  of  the  prescribed 
period,  with  certain  reservations  which  will  be  dealt  with 
later. 

In  both  the  instalment  and  annuity  methods  there  is  not 
any  question  of  the  rate  of  accumulation,  as  the  annual  repay- 
ments are  made  direct  to  the  lender,  and  tbere  is  not  therefore 
any  sinking  fund  set  aside. 

In  the  case  of  the  annuity  method  as  applied  to  the  repayment 
of  the  debt  of  a  local  authority,  the  lender  may,  or  may  not,  be 
able  to  reinvest  the  increasing  proportion  of  principal  included 
in  the  annual  instalment  paid  to  him,  at  the  calculated  rate 
which  he  receives  upon  his  investment,  but  this  does  not  enter 
into  the  calculation  in  any  way.  So  far  as  the  local  authority  is 
concerned  they  undertake  to  pay  to  the  lender  interest  at  the 
agreed  rate  for  such  period  only  during  which  they  have  the 
use  of  the  money.  As  regards  sinking  funds  relating  to  the  loan 
debt  of  commercial  and  financial  undertakings,  this  is  also 
generally  the  case,  but  the  purchaser  of  an  annuity  may  require 
that  the  annual  instalment  shall  be  fixed  at  such  an  amount  as 
will  yield  him  a  specified  rate  of  interest  upon  his  principal,  and 
at  the  same  time  enable  him  to  reinvest  the  annual  repayments 
of  principal  at  a  lower  rate  than  he  receives  as  interest,  in  order 
to  ro]ilncc  the  capital. 


THE    SINKING    FUND    METHOD  129 

With  regard  to  the  provision  in  the  Public  Health  Act  that 
the  annual  sum  set  apart  shall  be  sufficient,  after  iKiying  all 
expenses,  to  pay  oi!  the  money  borrowed  within  the  period 
sanctioned,  it  is  found  in  practice  that  the  expenses,  being  of 
uncertain  amount,  cannot  be  calculated  actuarially.  They  are 
therefore  omitted  from  the  calculation,  and  if  small  in  amount 
are  charged  direct  to  the  rate  or  revenue  account  as  and  when 
incurred.  Where  the  expenses  of  raising  the  loan  are  large  in 
amount,  as  is  the  case  when  the  loan  is  authorised  by  special 
Act  of  Parliament,  the  Act  generally  provides  that  the  cost  of 
obtaining  the  powers  shall  be  repaid  by  means  of  a  separate 
sinking  fund  to  mature  in  a  short  period,  generally  5  to  10  years. 
I  The  sinking  fund  method  is  the  one  now  generally  adopted 
by  all  local  authorities  for  the  annual  provision  for  redemption 
of  debt.  It  is  called  a  sinking  fund  when  it  relates  to  loans,  a 
loans  fund  when  it  relates  to  the  annual  provision  of  principal 
and  the  payment  of  dividends  on  stock,  and  a  redemption  fund 
when  it  relates  to  stock  issued  under  the  stock  regulations  of  the 
Local  Government  Board.  This  is  all  very  misleading  and 
confusing,  but  these  are  the  statutory  terms.  The  general  term 
sinking  fund,  with  some  distinguishing  word  added,  would 
better  describe  the  nature  of  the  fund  which  fulfils  the  same 
purpose  both  in  the  case  of  loans  and  stock.  The  sinking  fund 
relates  only  to  the  ultimate  repayment  of  principal  by  means 
of  an  equal  annual  sum  charged  against  the  year's  revenue  or 
rate,  such  annual  sum  being  set  aside  and  accumulated  by 
investment  in  outside  securities.  With  regard  to  the  interest 
payable  upon  the  loan,  it  is  obvious,  since  no  provision  is  made 
for  it  in  the  sinking  fund  instalment,  that  during  the  whole  of 
the  period  of  repayment  the  rate  or  revenue  account  of  each  year 
will  be  charged  with  interest  upon  the  full  amount  of  the 
original  loan,  and  this  notwithstanding  the  fact  that  part  of  the 
sinking  fund  may  have  been  applied  in  the  redemption  of  part 
of  the  loan  before  the  expiration  of  the  repayment  period. 

Since  the  interest  paid  upon  the  loan  is  quite  outside  the 
question  of  the  sinking  fund,  the  rate  of  accumulation  of  the 
fund  may,  and  generally  does,  differ  from  the  rate  of  interest 
payable  to  the  lender.  Section  234  (5)  of  the  Public  Health 
Act,  1875,  provides  that  the  local  authority  may  apply  the 
whole  or  any  part  of  the  sinking  fund  in  the  repayment  of  the 
debt,  but  if  they  do  so  they  must  pay  into  the  sinking  fund 
annually  a  sum  equivalent  to  the  interest  which  would  have 
been  produced  by  that  part  of  the  sinking  fund  so  applied. 
This  provision,  which  is  generally  inserted  in  all  general  and 


I30         REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

special  Acts,  is  absolutely  necessary.  The  sinking  fund  is 
calculated  to  accumulate  at  a  definite  rate  per  cent.,  and  if  any 
part  of  the  fund  be  used  to  repay  part  of  the  debt  the  fund  will 
be  deficient  to  that  amount,  and  will  lose  the  interest  upon  the 
portion  of  the  fund  so  applied.  This  provision  is  equal  to 
saying  that  any  such  application  of  the  sinking  fund  shall  be 
treated  as  an  investment  of  the  fund  as  if  it  had  been  actually 
invested  in  outside  securities. 

The  section  provides  that  the  local  authority  shall  pay  into 
the  sinking  fund  a  sum  equivalent  to  the  interest  which  would 
have  been  produced  by  that  part  of  the  fund  applied  towards 
the  redemption  of  debt.  But  in  practice  it  is  usual  to  estimate 
that  the  sinking  fund  will  accumulate  at  a  lower  rate  per  cent, 
than  the  interest  paid  upon  the  loan.  This  is  in  order  to 
provide  for  a  fall  in  the  rate  of  interest  obtainable  upon  first- 
class  investments,  and  it  results  in  a  larger  annual  instalment 
being  set  aside  than  would  be  the  case  if  the  sinking  fund 
were  calculated  to  accumulate  at  the  higher  rate  of  interest 
paid  upon  the  loan.  The  general  practice,  when  loans  are 
redeemed  out  of  the  sinking  fund,  is  to  pay  into  the  fund  the 
actual  amount  of  interest  previously  paid  to  the  loan  holders. 
Any  surplus  thus  arising  helps  to  make  up  the  deficiency  caused 
by  the  low  rate  of  interest  obtained  Avhen  part  of  the  sinking 
fund  is  in  the  bank  awaiting  investment,  as  often  happens. 

With  regard  to  the  investment  of  the  sinking  fund  until  it  is 
applied  in  the  redemption  of  debt,  it  was  until  recently  the 
practice  of  Parliament  and  also  of  the  Local  Government  Board 
to  require  that  it  should  be  invested  in  outside  securities,  but 
of  late  years  Parliament  has  given  power  under  special  Acts  to 
invest  the  sinking  funds  in  the  stocks  and  loans  of  the  same 
local  authority.  The  sinking  fund,  however,  cannot  be  invested 
in  any  other  department  of  the  same  authority  unless  that 
department  has  obtained  statutory  powers  to  borrow  the  amount, 
and  is  therefore  under  a  statutory  obligation  to  set  aside  out  of 
revenue  or  rate  a  sinking  fund  for  its  redemption. 

In  the  case  of  local  authorities  issuing  stock  at  par  which 
afterwards  commands  a  premium,  the  whole  of  the  cost  of  any 
part  of  the  stock  which  is  redeemed  at  a  premium  cannot  be 
taken  out  of  the  sinking  fund,  but  only  tlie  par  value  of  the 
stock,  the  premium  being  charged  to  the  rate  or  revenue  account 
at  the  time  the  stock  is  redeemed.  If  such  purchases  at  a 
premium  are  variable,  both  as  to  time  and  amount,  they  may 
be  dealt  with  by  means  of  a  su])plementary  sinking  fund  relating 
to  the  ]iremium  only,  in  such  a  manner  that  the  premium  is 


THE    SINKING    FUND    METHOD  131 

spread  equally  over  tlie  unexpired  period.  If  the  premium  is 
fixed  at  the  date  of  issue  of  the  stock  it  should  be  included  in 
the  original  sinking  fund  calculation,  but  if  the  stock  at  any 
time  commands  a  premium  beyond  this  amount  the  method  of 
providing  for  it  in  advance  will  be  more  difficult. 

The  Calculation  of  the  Annual  Instalment.  The  actual 
calculation  will  now  be  considered.  The  instalment  is  required 
to  be  set  aside  annually  and  accumulated  at  compound  interest 
in  order  to  provide  the  principal  sum  only,  and  the  question  of 
interest  upon  the  loan  does  not  enter  into  the  calculation. 
Under  these  conditions  it  would  appear  that  the  calculation  is 
much  simpler  than  in  the  annuity  method,  using  Tables  I  and 
III,  although  not  so  if  Table  Y  be  used.  The  question  to  be 
solved,  therefore,  is,  taking  as  before  a  loan  of  £1,000  repayable 
at  the  end  of  10  years  at  5  per  cent.,  "what  annuity  accumulated 
at  5  per  cent,  for  10  years  will  at  the  end  of  that  period  amount 
to  £1,000  "?  This  rate  per  cent,  is  the  rate  of  accumulation  of 
the  sinking  fund  and  not  the  rate  of  interest  payable  upon  the 
loan.  All  questions  involving  the  calculation  of  the  amount  of 
an  annuity  are  treated  by  the  formula  relating  to  Table  III, 
already  referred  to,  namely, 


M  =  A,(^l) 


the  actual  values  for  £1  per  annum  being  given  in  Table  III. 

The  sinking  fund  calculation  may  be  compared  with  that 
made  in  the  case  of  the  annuity  method,  Calculation  (XII)  3, 
in  which  the  instalment  was  required  to  provide  a  sum  equal  to 
the  "  amount  "  of  £1,000  accumulated  at  5  per  cent,  compound 
interest. 

In  this  case  the  instalment  has  to  provide  only  the  capital 
sum  of  £1,000  without  interest.  Consequently  if  the  actual 
loan  be  taken  instead  of  the  "  amount  "  of  the  same  sum  at 
the  end  of  the  period,  as  in  Calculation  (XII)  3,  the  required 
annual  instalment  will  be  obtained  for  the  reasons  given  in 
■discussing  Calculation  (XII)  1.  The  rule,  therefore,  to  find  the 
sinking  fund  instalment  is  :  — 

"  Divide  the  amount  of  the  loan  by  the  amount  of  £1  per 
annum  as  given  in  Table  III  for  the  required  number  of 
years  at  the  stated  rate  per  cent,  and  the  quotient  is  the 
required  annual  instalment.^' 


132  REPAYMENT    OF    LOCAL   AXD    OTHER   LOANS 

The  problem  resolves  itself  into  the  following  :  — 

If  £1  per  annum  in  10  years  at  5  per  cent,  -will,  at  the  end 

of  that  period,  amount  to  £12-5779,  what  annuity  will,  under 

the  same  conditions,  amount  to  £1,000? 

The  required  formula  is  obtained  by  transposing  the  formula 
relating  to  Table  III  as  follows  :  — 

A2/  =/ R^l  j  or  A2/  =  m(j^^^) 

and  the  calculation  will  be  made  upon  the  author's  standard 
form,  No.  3x,  by  the  three  methods  previously  referred  to. 

It  may  be  interesting  to  point  out  that  this  calculation  is  an 
example  of  how  the  use  of  a  formula  may  lead  to  the  discovery 
of  another  method  of  making  the  same  calculation.  It  will  be 
noticed  in  the  above  case  that  the  numerator  in  the  formula  is 
Mx7',  (which  means  that  £1,000  has  been  multiplied  by  0'05) 
and  the  result  divided  by  (RN_i).  But  £1,000  x  005  =  £50, 
which  is  the  interest  upon  £1,000  for  one  year  at  5  per  cent., 
and  therefore  that  an  alternative  rule  may  be  stated  as  follows  : 

"  To  ascertain  the  sinking  fund  instalment,  find  the  interest 
upon  the  amount  of  the  loan  for  one  year  at  the  sinking 
fund  rate  of  accumulation  (not  the  rate  of  interest  payable 
upon  the  loan)  and  divide  by  (E^'  — 1),  which  is  the  actual 
value  given  in  Table  I,  reduced  by  unity.'' 

This  rule  is  not  of  any  practical  advantage  over  those  given 
at  the  head  of  this  chapter,  and  will  not  therefore  be  further 
considered. 

The  Fixal  REPAYiiExx  of  the  Debt  by  the  Operatiox  of 
THE  Sinking  Fund.  The  following  statement  shows  the  final 
repayment  of  the  loan  by  the  operation  of  the  sinking  fund  and 
also  the  annual  payment  of  interest  upon  the  whole  of  the  loan 
until  the  end  of  the  prescribed  period  when  the  accumulation  of 
the  fund  is  equal  to  the  amount  of  the  loan  which  is  then 
repaid,  the  fund  exhausted,  and  the  annual  contributions  cease. 

This  statement  shows  that  the  fund  is  increased  annually 
by  the  instalment  provided  out  of  revenue  or  rate  and  by  the 
income  received  upon  the  investment  of  previous  instalments. 
This  income  from  investments  is  the  amount  which  the  lender, 
under  the  annuity  method,  would  have  obtained  if  he  had  taken 
out  of  each  annual  instalment  of  £12951  paid  to  him  the  sum 
of  £50   bv  wav  of   interest   upon   his   loan,    and   invested   the 


THE    vSINKING    FUND    METHOD  I33 

reiiiaiuiug-  £79-51  and  the  subsequent  accumulations  at  5  per 
cent,  annually  to  provide  his  capital  at  the  end  of  the  term. 
He  would  by  this  means  obtain  a  more  regular  income  than  by 
treating  as  income  the  interest  shown  in  the  tables  relating  to 
the  annuity  method,  which  decreases  year  by  year.  It  will 
further  be  noticed  that  the  interest  charged  to  the  revenue  or 
rate  account  under  the  annuity  method,  as  shown  in  the  table 
relating  to  that  method,  added  to  the  income  received  from 
investments,  as  shown  in  the  table  relating  to  the  sinking  fund 
method,  are  together  equal  in  each  year  to  £50,  which  is  the 
interest  paid  to  the  lender  annually  under  the  sinking  fund 
method.     See  Statement  XIII.   A.,  page  139. 

If,  therefore,  the  lender,  under  the  annuity  method,  requires 
to  equalise  his  annual  income,  he  may  do  so  by  setting  aside  an 
equal  annual  sum  out  of  the  instalment  and  accumulating  it  as 
a  sinking  fund  to  provide  his  capital. 

This  mode  of  equalising  the  income  might  be  adopted  by 
trustees  and  executors  with  the  object  of  securing  a  fixed  income 
for  a  tenant  for  life,  but  will  apply  only  to  an  annuity  for  a 
fixed  term.  The  above  argument  is,  however,  subject  to  the 
reservation  that  the  lender  may  not  be  able,  year  by  year,  to 
reinvest  the  periodical  repayments  of  principal  to  yield  the  rate 
per  cent,  upon  which  the  annual  instalment  Avas  based. 

Comparison  of  the  Theee  Methods.  It  is  now  possible  to 
compare  the  repayment  of  loans  by  instalment,  annuity  and 
sinking  fund  methods,  as  above  described,  and  this  will  be 
done  from  the  standpoints  both  of  the  lender  and  borrower  by 
means  of  the  following  statement  (XIII.  B.,  page  140. 

In  Chapter  XI,  the  instalment  method  has  been  compared 
with  the  annuity  method,  and  it  is  interesting  to  compare  the 
annuity  method  with  the  sinking  fund  method.  In  each  case 
the  annual  instalment  is  ascertained  by  dividing  a  definite  sum 
by  the  same  accumulated  amount  of  an  annuity  of  £1  for  10 
years  at  5  per  cent.,  but  in  the  case  of  the  annuity  method 
the  amount  so  divided  is  the  amount  of  the  principal  sum 
accumulated  at  compound  interest,  whilst  in  the  sinking  fund 
method  the  amount  so  divided  is  the  principal  sum  itself 
without  accumulations.  This  is  owing  to  the  fact  that  the 
annual  instalment  in  the  case  of  the  annuity  method  includes 
interest,  whereas  the  annual  instalment  in  the  case  of  the 
sinking  fund  relates  to  the  principal  sum  only.  The  annual 
instalment  in  the  sinking  fund  method,  therefore,  is  smaller 
than  in  the  case  of  the  annuity  method. 


134    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

By    tlie    annuity   method,    Calculation    (XII)  3,    the 

instalment  of  principal  and  interest  is £129'51 

By    the    sinking    fund    method    the    instalment    of 

principal  only  is       £79"51 

The  difference  being-  one  year's  interest  on  £1000  at 

5  per  cent \     £5000 

Under  the  sinking  fund  method,  therefore,  the  total  annual 
charge  to  revenue  or  rate  in  respect  of  principal  and  interest  is 
exactly  equal  year  by  year  to  the  total  annual  charge  under  the 
annuity  method,  viz.  £129'51  in  each  case. 

This  has  already  been  referred  to  in  discussing  the  annuity 
method  in  Chapter  XII. 

With  regard  to  the  instalment  method  the  total  annual 
charge  to  revenue  or  rate  account  in  respect  of  principal  and 
interest  is  greater  in  the  earlier  years  and  is  gradually  reduced 
from  £150  to  £105  in  ten  years.  The  relative  merits  of  the 
annuity  method  and  the  sinking  fund  method  as  regards  the 
annual  incidence  of  local  taxation  are  equal  and  are  more 
equitable  than  the  instalment  method.  As  regards  the  investor, 
under  the  instalment  method  he  receives  a  decreasing  annual 
payment  made  up  of  a  constant  amount  of  principal  and  a 
decreasing  amount  of  interest ;  but  he  has  definite  knowledge  of 
how  much  is  interest  and  how  much  is  principal.  Under  the 
annuity  method  he  receives  an  equal  annual  payment  made  up 
of  an  increasing  amount  of  principal  and  a  decreasing  amount  of 
interest;  but  without  an  elaborate  calculation  he  is  unable  to 
apportion  the  amount  paid  to  him  between  capital  and  income. 
Under  both  the  instalment  and  the  annuity  methods  the 
investor  receives  annual  sums  in  respect  of  his  capital  which 
he  has  to  reinvest  in  small  amounts. 

Comparing  the  sinking  fund  method,  on  the  one  hand,  with 
the  instalment  and  annuity  methods  on  the  other,  from  the 
point  of  view  of  the  investor,  it  will  be  seen  that  under  the 
sinking  fund  method  he  receives  each  year  an  equal  amount  by 
way  of  interest  upon  his  money,  and  has  the  further  advantage 
of  a  permanent  investment  of  the  whole  of  his  capital  for  a 
definite  long;  term.  If  he  wishes  to  realise  he  has  a  definite 
security  to  place  upon  the  market  either  to  be  bought  by  some 
other  investor  or  to  be  redeemed  by  the  local  authority  out  of 
the  sinking  fund.  lender  the  sinking  fund  method  he  has  to 
run  the  risk  of  a  fall  in  the  market  value  in  the  case  of  a  loan 


THE    SINKING    FUND    METHOD  135 

raised  l>v  tlie  issue  of  stock;  but,  on  tlie  other  hand,  he  may- 
realise  a  profit.  Summing  up  the  respective  merits  of  the  various 
methods  of  repayment  of  the  debt  of  local  authorities,  it  may 
fairly  be  concluded  that  the  accumulating  sinking  fund  method 
is  by  far  the  best.  It  bears  equally  upon  the  taxation  or 
revenue  of  each  year  of  the  repayment  period;  and  as  regards 
the  investor,  it  is  at  once  more  convenient  and  more  equitable 
than  either  of  the  other  two  methods. 


The  Non-Accumulating  Sinking  Fund.  Up  to  this  point 
the  enquiry  has  been  limited  to  accumulating  sinking  funds 
similar  to  the  one  prescribed  in  the  Public  Health  Act,  1875. 
The  principal  feature  of  such  a  fund  is  the  provision  out  of 
revenue  or  rate  of  an  equal  annual  instalment  to  be  set  aside 
and  accumulated  for  a  prescribed  period  at  a  rate  per  cent,  to  be 
fixed  in  anticipation,  with  as  near  approach  to  accuracy  as  can 
be  obtained.  In  the  case  of  loans  with  long  repayment  periods 
this  is  very  difficult,  and  it  therefore  becomes  necessary  to 
compare  the  actual  amount  in  the  fund  periodically  with  the 
calculated  amount  which  should  be  in  the  fund  as  shown  by  the 
pro  forma  account.  Any  surplus  or  deficiency  in  an  accumulat- 
ing fund  should  be  credited  to,  or  charged  against,  the  revenue 
or  rate  account  of  each  year,  but  this  entails  considerable 
labour,  and  it  is  one  of  the  objects  of  the  non-accumulating 
sinking  fund  to  avoid  this  by  providing  an  automatic  accurate 
accumulation  of  the  fund  irrespective  of  the  rate  of  income 
received  on  the  investments  representing  the  fund.  The  basis 
of  the  method  is  the  instalment  system  discussed  in  Chapter  XI, 
where  each  year  a  definite  sum  is  charged  to  the  revenue  or  rate 
account  and  repaid  to  the  lender.  The  annual  instalment  in 
the  case  of  the  non-accumulating  sinking  fund  is  calculated 
precisely  as  in  the  instalment  method,  namely,  by  dividing  the 
amount  of  the  loan  by  the  number  of  years  in  the  repayment 
period.  But  in  this  case  the  annual  instalment  of  principal  is 
not  repaid  to  the  lender,  but  is  invested  by  the  local  authority 
in  order  to  provide  the  amount  of  the  loan  at  the  end  of  the 
period.  Since  an  equal  amount  is  added  to  the  fund  year  by 
year  it  requires  merely  an  arithmetical  calculation  to  ascertain 
the  amount  which  should  be  in  the  fund  at  any  time.  Seeing 
that  the  total  amount  of  the  loan  is  provided  by  the  actual  equal 
annual  charges  to  revenue  or  rate,  it  is  obvious  that  the  income 
arising  from  the  investments  rej^resenting  the  fund  need  not  be 
added  to  the  fund.     On  comparing  the  actual  instalments  only. 


136    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

under  tlie  instalment  metliod  in  Chapter  XI,  with  tliose  under 
the  accumulating  sinking  fund  method  in  this  chapter,  it -will  he 
seen  that  the  charge  to  revenue  or  rate  under  the  instalment 
method  is  greater  than  under  the  accumulating  sinking  fund 
method,  consequently  in  the  case  of  a  non-accumulating  fund 
tlie  income  to  arise  from  the  investments  may  be  credited  to  the 
rate  or  revenue  account  to  which  the  annual  instalment  of 
principal  has  been  debited.  The  excess  of  the  original  annual 
instalment  in  the  non-accumulating  fund  over  the  instalment  m 
the  accumulating  fund  Avill  not  be  compensated  by  the  reduction 
therein  due  to  the  income  received  from  the  investment  of  the 
fund,  because  such  resulting  income  will  be  small  during  the 
earlier  years ;  and  an  equality  in  the  annual  burden  will  not  be 
reached  until  the  end  of  the  fifth  year  out  of  ten.  The  first  four 
years  will  therefore  bear  an  additional  burden,  and  the  last  five 
years  will  be  relieved,  as  compared  with  the  annual  incidence 
under  the  accumulating  fund,  in  a  similar  manner  to  the  instal- 
ment method.  As  regards  the  ratepayer,  the  non-accumulating 
fund  will  have  all  the  disadvantages  of  the  instalment  method 
already  pointed  out.  The  lender,  on  the  contrary,  will  be  in  a 
better  position,  since  he  obtains  a  permanent  investment  and  is 
relieved  of  tlie  periodical  reinvestment  of  small  amounts  of 
capital.  The  actual  method  by  which  the  local  authority 
provides  the  sinking  fund  has  not  any  particular  interest  to  him. 
As  between  the  instalment  and  annuity  methods,  on  the  one 
hand,  and  the  two  sinking  fund  methods  on  the  other,  the  only 
difference  is  the  date  at  which  he  shall  be  repaid,  and  he  invests 
in  the  particular  loan  which  best  meets  his  requirements.  Under 
the  two  periodical  repayment  methods  the  lender  may  be  said  to 
keep  his  own  sinking  fund,  whereas  in  both  sinking  fund 
methods  the  local  authority  does  this  for  him.  There  are  not 
any  mathematical  principles  involved  in  the  non-accumulating 
fund,  but  it  is  merely  an  arithmetical  one.  The  following  table 
shows  the  final  repayment  of  the  loan  by  the  operation  of  the 
fund.  In  order  that  it  may  be  compared  with  the  accumulating 
sinking  fund  it  has  been  assumed  that  the  investments  yield 
6  per  cent,  per  annum,  and  that  no  part  of  tlie  fund  is  ap])lied  in 
I'edemptinn  of  debt  during  the  period  :  — 


THE    SINKING    FUND    METHOD  137 

STATEMENT  XIII.  C. 

The  Repayment  of  the  Debt  of  Local  Altthoeities. 
The  Non-Accumulatixg  Sinking  Fund. 

Showing  the  repajanent  of  a  loan  of  £1,000,  at  the  end  of 
10  years  by  an  equal  annual  instalment  of  principal,  to  be 
set  aside  and  invested  as  a  sinking  fund,  the  annual  income 
upon  the  investments  being  applied  in  reduction  of  subse- 
quent instalments.     Interest  at  5  per  cent. 

Net  Total 

Annual        Deduct   charge  to  charge  to     Amount 


^ear 
end 

Instal- 
ment 

Income 
received 

revenue 
or  rate 

Interest 
on  loan 

revenue 
or  rate 

in 
fund 

Year 
end 

1 

100 

— 

100 

50 

150 

100 

1 

2 

100 

5 

95 

50 

145 

200 

2 

3 

100 

10 

90 

50 

140 

300 

3 

4 

100 

15 

85 

50 

135 

400 

4 

5 

100 

20 

80 

50 

130 

500 

5 

6 

100 

25 

75 

50 

125 

600 

6 

7 

100 

30 

70 

50 

120 

700 

7 

8 

100 

35 

65 

50 

115 

800 

8 

9 

100 

40 

60 

50 

110 

900 

9 

10  100  45  55  50  105         1000         10 

The  clauses  authorising  the  non-accumulating  sinking  fund 
contain  the  usual  permission  to  apply  part  of  the  fund  in 
redemption  of  debt;  but  in  this  case  there  is  not  any  necessity 
to  have  regard  to  the  interest  which  would  have  been  received  in 
respect  of  the  part  of  the  fund  so  applied,  because,  although  the 
income  received  from  the  investments  will  be  smaller  in  conse- 
quence of  such  application,  yet  the  interest  payable  upon  the 
loan  will  be  correspondingly  reduced,  and  there  will  not  there- 
fore be  any  alteration  in  the  combined  charge  for  interest  and 
redemption.  There  is  another  matter  which  may  properly  be 
considered  to  the  advantage  of  the  non-accumulating  fund,  if 
the  greater  burden  imposed  during  the  earlier  years  is  not  fatal 
to  its  adoption.  This  affects  the  possible  and  very  probable 
variation  in  the  rate  of  income  to  be  received  from  the  invest- 
ments representing  the  fund.  In  the  ease  of  the  accumulating 
fund,  as  already  pointed  out,  it  very  rarely  happens  that  the 


138  REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 

fund  increases  in  accordance  with  the  calculated  amount,  thus 
rendering  it  necessary  to  make  frequent  adjustments  through 
the  revenue  or  rate  account.  As  will  be  shown  in  later  chapters, 
the  variation  in  the  rate  per  cent.,  whether  of  income  from 
investments  or  of  accumulation,  gives  rise  to  many  of  the 
problems  which  have  to  be  dealt  with.  The  non-accumulating 
fund  has  this  advantage,  that  any  variation  in  the  rate  per  cent, 
is  at  once  automatically  adjusted,  seeing  that  if  from  any  cause 
there  is  a  fall  in  the  rate  of  income  from  investments  there  is  a 
corresponding  increase  in  the  charge  to  revenue  or  rate  due  to 
the  decreased  relief  to  subsequent  annual  instalments  afforded 
by  the  amount  of  income  received  from  the  investments. 

In  the  case  of  a  non-accumulating  fund  the  annual  instal- 
ment of  the  full  amount  would  be  credited  each  year  to  the 
sinking  fund  account  and  not  debited  direct  to  the  revenue  or 
rate  account,  but  to  an  intermediate  account  which  might  be 
called  the  "  non-accumulating  sinking  fund  suspense  account." 
This  suspense  account  Avould  be  credited  with  the  income  received 
from  the  investments  representing  the  fund,  and  with  interest 
allowed  by  the  bank,  if  any,  whether  any  part  of  the  fund  had 
been  applied  in  redemption  of  debt  or  not,  and  it  would  be 
debited  with  the  interest  actually  paid  or  accrued  on  the 
outstanding  loan.  The  balance  remaining  to  the  debit  of  this 
suspense  account  would  then  be  charged  to  the  revenue  or  rate 
account  and  would  represent  the  total  charge  against  the  year  in 
respect  of  loan  indebtedness.  The  accounts  of  an  accumulating 
sinking  fund  might  be  kept  in  a  similar  manner.  A  "  sinking 
fund  interest  suspense  account "  would  be  opened,  and  the 
sinking  fund  account  would  be  credited  and  the  suspense  interest 
account  debited  with  the  actual  amount  of  interest  which  should 
yearly  accrue  to  the  fund  as  shown  by  the  pro  forma  account. 
The  suspense  interest  account  would  be  credited  with  the  actual 
income  received  from  the  investments,  and  bank  interest;  and 
the  balance,  either  debit  or  credit,  but  generally  debit,  would  be 
closed  by  transfer  to  the  revenue  or  rate  account.  By  this 
means  the  sinking  fund  accounts  Avould  always  stand  at  their 
proper  calculated  amounts,  any  reduction  in  income  would 
immediately  become  apparent  and  there  would  be  no  possibility 
of  the  gradual  accumulation  of  many  small  deficiencies  requiring 
at  some  future  time  considerable  correction  by  an  increased 
annual  instalment. 


THE    SINKING    FUND    METHOD 


139 


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140    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


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THE    SINKING    FUND    METHOD 


141 


Calculation  (XIII)  1. 

Standard  Calculation  Form,  No.  3x. 

To  find  the  annual  sinking  fund  instalment.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
as  a  sinking  fund  at  5  per  cent,  to  provide  £1,000  at  the 
end  of  10  years. 


A)     By  Formula.          A(/=  M  (j^y'l^)  R^^le  1,  Chapter  XIII. 


Log 
R^— 1 


'Log.  Ratio 

Multiply  Log.  R  by 

Convert  Log. 

to  ordinary  number 
deduct  unity 


R 

N 


Log  of  this  is 
og.  Amount 
add  Loo^,  r 


105 
10 


00211893 
10 


Los".  Amount  of  Loan 


deduct  Log.  (R^  - 1)  above 


RN 

(1-05)10 

0-2118930 

R^' 
-1 

1-6289 
1- 

RN-1 

0-6289 

1-7985779 

M 

r 

1000 
-05 

3- 
2-6989700 

Mr 

RN-1 

1-6989700 
1-7985779 

A?/ 

1-9003921 

Required  annual  instalment,   £79-5046. 
M 


(B)     By  Table  III.         A^  =  RN-1  Rule  2,  Chapter  XIII. 


Log.  Amount  of  Loan     M 
Table  III.     10  years,  5  per  cent 
Amount  of  £1  per  annum 
deduct  Log. 


1000 

R^-1     12-5779 
r 


1-0996079 


A?y 


1-9003921 


Required  annual  instalment,   £79-5046. 


(C j     By  Thoman's  Table.     A^  =  ^^(^)  ^^^^^  ^'  Cliapter  XIII. 
5  per  cent.,  10  years. 


Log.  Amount  of  Loan 
add  Log.  a" 

M 

a^ 

1000 

3- 
9-1122851 

deduct  Log.  R^  in 
Table +10 

Ma« 

RN 

121122851 
10-2118930 

A.v 

1-9003921 

Required  annual  instalment,   £795046. 


Section  III. 
Sinking  Fund  Problems. 


The  Annual    Instalment. 


vSINKING    FUND    PROBLEMS  145 


CHAPTER  XIY. 
SINKING  FUND  PEOBLEMS. 

RELATING   TO : 

(U  The  Amount  in  the  Fund. 

(2)  The  E ate  per  cent  :  — 

(a)  of  income  to  be  received  upon  the  present  invest- 
ments representing  the  fund  ; 
(6)  the  future  rate  of  accumulation. 

(3)  The  Eedemption  Period. 

(4)  The   rate   per   cent,    and   the   redemption   period   in 

combination. 

Definition  of  terms  : 

1.  The  present  inve.stments. 

2.  The  annual  increment. 

The  Pro  forma  Account.  Having  discussed  the  several 
alternative  methods  of  repayment  of  loan  debt  by  local  authori- 
ties laid  doAvn  by  statute,  and  having  described  the  methods 
of  finding  the  annual  sums  to  be  set  aside  for  that  purpose  out 
of  revenue  or  rate,  the  subject  M'ill  now  be  considered  in  its 
practical  aspect.  Most  of  these  transactions  extend  over  very 
long  periods  and  all  trace  of  the  original  calculation  is  often 
lost.  It  is  therefore  advisable  in  all  cases  involving  a  periodical 
provision  for  repayment  by  means  of  a  sinking  fund  to  prepare 
at  the  outset  a  pro  forma  account  showing  how  the  calculated 
annual  instalment  should  work  out  during  the  whole  of  the 
period . 

The  Local  Government  Board  auditors  in  many  cases  require 
this  to  be  done  in  respect  of  all  loans  coming  under  their 
supervision,  and  it  is  a  practice  to  be  commended  and  followed. 
Such  a  pro  forma  account  enables  a  comparison  to  be  made 
annually  between  the  actual  and  the  calculated  working  out  of 
the  fund  so  that  any  discrepancy  may  be  immediately  set  right. 
Especially  does  this  apply  to  a  deficiency  caused  by  part  of  the 
sinking  fund  lying  uninvested  in  the  bank  and  earning  less  than 


146         REPAYMENT   OF   LOCAL   AND    OTHER   LOANvS 

the  calculated  rate  per  cent,  of  accumulation  or  due  to  a  general 
decrease  in  such  rate.  Any  such  deficiency  will  be  of  small 
amount  in  any  one  year,  and  may  be  charged  against  the 
revenue  or  rate  account  of  the  particular  year,  so  keeping  the 
sinking  fund  up  to  the  proper  amount.  But  cases  have  arisen 
in  which  this  has  not  been  done,  and  from  the  above,  and  other 
causes,  the  amounts  in  the  sinking  funds  have  been  seriously 
deficient.  In  such  cases  it  becomes  necessary  to  ascertain  the 
proper  amount  which  would  have  been  in  the  fund  if  the 
original  anticipations  had  been  realised.  This  is  a  contingency 
which  may  arise  in  the  case  of  a  local  authority,  and  there  are 
other  questions  with  regard  to  sinking  funds  which,  although 
not  affecting  local  authorities,  yet  are  very  important  in  con- 
nection with  the  sinking  funds  of  commercial  and  financial 
undertakings. 

Nature  of  Problems.  In  dealing  with  all  cases  of  adjust- 
ment of  a  sinking  fund  it  will  be  necessary  to  refer  continually 
to  the  present  state  of  the  fund  as  the  basis  upon  which  all  such 
adjustments  are  made,  and  later,  when  dealing  with  other 
problems,  it  will  be  seen  that  the  present  position  of  the  fund 
plays  an  equally  important  part.  Such  questions  will  be 
considered  later,  and  will  comprise  :  — 

(1)  A  deficiency  in  the  fund.  Chapters  XY,  XYI. 

(2)  A  surplus  in  the  fund.  Chapters  XYII,  XYIII. 

(3)  A  variation  in  the  rate  per  cent,  at  which  the  fund  was 

originally  expected  to  accumulate.         Chapter  XIX,  etc. 

(4)  A  variation  in  the  rate  of  income  to  be  yielded  by  the 

investments  representing  the  fund. 

Chapters  XX,  XXYII. 

(5)  A  variation  in  the  repayment  period.  Chapter  XXIY. 

(6)  A  variation  in  the  repayment  period  accompanied  by  a 

variation  in  the  rate  of  accumulation.       Chapter  XXYI. 

Any  or  all  of  the  above  contingencies  may  have  to  be  taken 
into  account  in  an  adjustment,  and  as  they  arise  only  after  the 
fund  has  been  in  operation  for  part  of  the  original  repayment 
period,  it  is  important  to  ascertain  exactly  the  position  of  the 
fund  at  the  time  the  adjustment  is  required  to  be  made.  It  is 
generally  the  case  with  the  sinking  funds  of  local  authorities 
that  the  amount  standing  to  the  credit  of  the  fund  is  required 


SINKING    FUND    PROBLEMS  I47 

to  be  invested  in  specific  outside  securities  allocated  to  the  fund, 
or,  wliicli  is  the  same  in  effect,  shall  have  been  applied  in  part 
repayment  of  the  original  loan.  In  the  case  of  commercial  and 
financial  undertakings  it  is  usual  to  impose  the  obligation  of 
such  outside  investment  in  order  to  ensure  that  the  original 
purpose  of  the  fund  shall  be  carried  out,  and  that  the  amount  in 
the  fund  shall  be  actually  available  for  the  repayment  of  the 
debt  at  the  end  of  the  period.  Any  enquiry  therefore  into  the 
adequacy  or  otherwise  of  the  amount  in  the  fund  at  any  time 
will  properly  include,  not  only  the  value  of  the  investments 
representing  the  fund  at  the  present  time,  but  also  an  enquiry 
as  to  the  probable  value  at  the  end  of  the  repayment  period. 
It  will  be  necessary  to  ascertain  whether  they  are  yielding  or 
are  likely  to  continue  to  yield  a  retvirn  by  way  of  income  equal 
to  or  differing  from  the  calculated  rate  percent,  of  accumulation. 
In  the  following  chapters,  treating  of  the  above  possible  causes 
of  variation,  as  far  as  possible  for  purposes  of  convenience  and 
comparison,  the  position  of  an  imaginary  sinking  fund  will  be 
ascertained  at  the  end  of  the  12th  year  of  an  original  period 
of  25  years,  and  the  position  of  the  fund  will  be  shown  at  that 
time,  when  the  enquiry  and  any  subsequent  rectification  is 
made,  in  the  following  terms,  viz.  :  — 

(1)  The.   vahie   of  the  present  investments   representing  the 

amount  in  the  fund. 

(2)  The  present  annual  increment  at  the  time  the  enquiry  is 

made,  and  before  the  rectification  to  meet  the  new  condi- 
tions. 

Present  Investments.  The  term  "  present  investments  " 
will  be  used  to  denote  the  value  of  the  investments  representing 
the  amount  which  actually  stands  to  the  credit  of  the  fund  and 
not  the  amount  which  should  so  stand  by  calculation  at  the 
original  rate  of  accumulation  as  shown  by  the  pro  forma 
account.  In  fixing  the  precise  market  value  regard  should  be 
had  to  the  probability  of  the  individual  investments  ultimately 
yielding  the  original  cost  price,  and  if  any  fall  in  value  has 
occurred,  or  is  likely  to  occur,  it  should  as  far  as  possible  be 
included  in  the  adjustment.  In  dealing  with  a  surplus  or  a 
deficiency  in  the  fund,  any  actual  change  in  value  should  be 
taken  into  account  in  calculating  the  amended  annual  instal- 
ment ;  but  where  the  problem  concerns  the  period  of  repayment 
or  the  rate  of  accumulation,  and  especially  if  the  fund  has  a 
long  unexpired   period   to  run,   it   is   hardly  possible  to   make 


I4S    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

any  exact  forecast  of  tlie  future  value  of  the  iuvestments,  or  of 
the  future  rate  of  income  to  be  receiTed  therefrom,  and  this 
should  be  provided  for  by  making  an  allowance  when  deciding 
upon  the  amended  rate  of  accumulation,  namely,  by  taking  it  at 
a  slightly  lower  rate  than  would  otherwise  be  sufficient.  In  the 
whole  of  the  following  examples,  except  a  deficiency  or  a 
surplus  in  the  fund,  it  will  be  assumed  that  the  fund  stands  at 
the  exact  amount  shown  by  the  original  calculation ;  and, 
further,  that  the  various  investments  representing  the  fund  are 
each  worth  now  the  exact  amount  paid  for  them,  and  will  be 
so  at  the  end  of  the  period.  This  will  sufiiciently  explain 
without  further  reference  the  meaning  attached  to  the  term 
"  present  investments  "  in  the  following  pages. 

The  Annual  Increment.  With  regard  to  the  annual 
increment,  it  will  be  seen,  on  considering  the  sinking  fund  at 
its  inception,  that  there  is  then  only  one  factor  to  deal  with, 
namely,  the  repayment  of  a  definite  loan  (or  the  provision  of  a 
definite  sum)  at  the  end  of  a  stated  number  of  years.  This 
term  will  be  referred  to  in  the  following  pages  as  the  "  period  of 
repayment  or  redemption,"  and  in  order  to  make  the  adjustment 
it  is  necessary  to  fix  an  average  rate  per  cent,  at  which  the 
future  payments  to  the  fund  may  reasonably  be  expected  to 
accumulate  by  subsequent  investment.  It  is  very  difficult,  if 
not  impossible,  to  do  this  correcth'  in  the  case  of  a  fund  having 
a  long  period  of  repayment,  and  the  practice  generally  is  to 
assume  a  rate  of  accumulation  slightly  lower  than  the  rate  of 
interest  payable  to  the  loauholders.  This  will  alloAv  for  a  fall 
in  the  accumulation  rate  owing  to  fluctuations  of  the  money 
market  or  for  a  deficiency  in  the  income  of  the  fund  caused  by 
delay  in  finding  an  investment  which  leaves  money  idle  in  the 
bank,  earning  only  a  low  rate  of  interest.  If  the  annual 
deficiency  in  the  income  of  the  fund  or  any  annual  surplus  be 
small  it  should  be  rectified,  as  and  when  it  arises,  by  adjusting 
it  by  means  of  the  revenue  or  rate  account,  but  if  the  annual 
deficiency  or  surplus  be  large,  it  is  better  to  adjust  the  annual 
instalment  immediately  in  the  manner  to  be  described  later 
under  the  head  of  variation  in  the  rate  of  accumulation.  Having 
fixed  the  future  estimated  rate  of  accumulation,  the  calculation 
is  made  in  the  manner  shown  in  Calculation  (XIII)  1,  to 
ascertain  the  annual  instalment  to  be  set  aside  each  year  to 
accumulate  at  the  estimated  rate.  This  annual  instalment  thus 
becomes  the  annual  increment  during  the  first  year,  but  after 
tl)e  first  instalment  has  been  invested  another  factor  is  introduced 


SINKING    FUND    PROBLEMS  i49 

into  the  annual  increment,  namely,  the  income  from  the  invest- 
ments representing  the  fund. 

It  is  not  often  that  any  question  affecting  the  adequacy  of 
the  amount  in  the  fund  arises  during  the  earlier  years  of  the 
repayment   period.     Generally    it    is    much    later,    and   in    the 
following  examples  it  has  been  taken  as  the   12th  year  of   a 
period  of  25  years.     By  this  time  the  fund  will  have  amounted 
to  a  large  proportion  of  the  total  sum  to  be  ultimately  provided, 
and  the  accruing  annual  income  from  investments  will  (with  a 
3i  per  cent,  rate  of  accumulation)   be  about  one-half   of  the 
original  annual  instalment.     Any  adjustment  of  the  fund  at 
the  end  of  the  12th  year  will  therefore  depend  largely  upon  the 
future  rate  of  income  to  be  yielded  by  the  present  investments 
representing  the  fund.     And  this  adjustment  may  actually  be 
rendered  necessary  by  a  fall  in  the  rate  of  income  yielded  by 
the  present  investments,    occurring   at   a  time  when  the   rate 
yielded  by  other  investments  of  all  kinds  is  also  falling.     If  the 
original  rate  of  accumulation  be  likely  to  be  maintained   in 
spite  of  a  fall  in  the  income  received  from  the  present  invest- 
ments, there  is  not  any  need,  as  shown  in  Chapter  XX  (variation 
B,  in  the  rate  per  cent,  of  income)  to  make  any  adjustment  by 
calculation  in  the  annual  instalment.     All  that  is  required  is  to 
take  an  additional  annual  sum  out  of  revenue  or  rate,  equal  to 
the  amount  of  the  reduction  in  the  future  annual  income  to  be 
received    from    the    present    investments,    and    the    fund    will 
continue  to  accumulate  as  originally  calculated.     But  where,  as 
in    Chapter    XXI    (variation    C    in    the    rate   per    cent.)    it    is 
necessary  at  the  same  time  to  provide  for  a  reduction  in  the 
rate    of    income    from    the    present    investments    as    well    as    a 
reduction   in  the  rate  of   accumulation,   the  problem  becomes 
more  complicated  because  there  are  then  two  different  rates  per 
cent,  acting  upon  two  diff'erent  factors.     The  rate  of  income 
upon  the   present   investments  has  no  relation  to  the   annual 
instalment  provided  out  of  revenue  or  rate  which  is  acted  upon 
by  the  accumulation  rate  only.     But  the  actual  amount  (if  not 
the  rate  per  cent.)  of  the  income  from  investments  is  also  acted 
upon   by   the   accumulation   rate,    and    it   is   possible   to   state 
definitely  the  annual  sum  which  will  be  received  in  respect  of 
such  income.     Consequently,   the  difficulty  attending  the  two 
rates  per  cent,  may  be  avoided  by  treating  the  future  income 
from  the  present  investments  as  an  annuity  certain  which  will 
continue  to  be  received   during   the   whole   of   the   unexpired 
portion  of  the  repayment  period  in  exactly  the  same  way  as  the 
original  annual  instalment  will  continue  to  be  set  aside  out  of 


150    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

revenue  or  rate.  These  two  annual  factors  tog-ether  will  be 
considered  as  constituting  the  future  annual  increment  to  be 
included  as  an  asset  in  the  adjustments,  and  to  be  supplemented, 
as  will  be  seen  later,  by  any  additional  annual  instalment  (to  be 
provided  out  of  revenue  or  rate)  which  may  be  found  necessary 
to  make  up  for  the  decrease  in  the  income  from  the  present 
investments,  and  also  any  further  deficiency  caused  by  a 
reduction  in  the  rate  of  accumulation.  This  supplemented 
annual  increment  will  be  referred  to  later  as  the  future  or 
amended  annual  increment,  as  defined  in  Chapter  XXII. 
Although  in  the  examples  which  will  be  considered  later  a 
reduction  in  the  rate  of  income  from  investments  will  be 
assumed,  it  is  quite  possible  that  there  may  be  an  increase  in 
the  rate  of  income,  which  would  have  the  effect  of  reducing  the 
original  annual  instalment  instead  of  increasing  it.  It  rarely 
happens,  however,  that  there  is  an  increase  in  the  rate  of 
accumulation.  It  is  unwise  to  predict  a  change  which  will 
have  the  effect  of  relieving  the  present  revenue  or  rate  account 
to  the  possible  detriment  of  future  years,  and  if  any  surplus  in 
the  fund  arises  in  this  way  it  is  usually  dealt  with  at  the  time. 
The  above  remarks  will  explain  the  reason  for  the  methods 
adopted  later  of  showing  the  position  of  the  fund  at  the  end 
the  12th  year  when  dealing  with  variations  in  the  rates  per  cent, 
of  accumulation  and  income  which  dift'er  from  the  methods 
adopted  to  show  the  position  at  the  end  of  a  similar  period  when 
dealing  with  a  deficiency  or  a  surplus  in  the  amount  in  the 
fund  or  with  variations  in  the  period  of  repayment  without  any 
variation  in  the  rate  per  cent,  either  of  income  or  of  accumula- 
tion. In  both  the  latter  cases  (see  Statements  XV.  B.  and 
XXIV.  A.),  which  do  not  involve  any  variation  in  the  rate 
of  accumulation  or  in  the  rate  of  income,  the  assets  of  the  fund 
include  the  accumulated  amount  (using  the  term  as  in  Table  I) 
of  the  value  of  the  present  investments  at  the  end  of  the 
respective  re])ayment  periods.  This  amount  includes  tlie 
present  value  of  the  investments  (£916-3  and  £9932" 74)  and 
the  accruing  compound  interest,  because  they  both  accumulate 
at  the  same  rate  which  is  the  same  as  the  rate  of  income  upon 
the  investments. 

But  m  problems  involving  a  variation  in  the  rate  per  cent, 
of  accumulation,  without  any  variation  in  the  rate  of  income 
from  investments  (as  in  Statement  XIX.  A.)  it  is  necessary 
to  find  the  future  amount  of  the  present  investments  by  two 
calculations  because  whilst  the  present  investments  continue 
to  yield  3^  per  cent,  per  annum,  the  income  so  yielded  accumu- 


SINKING    FUND    PROBLEMS  151 

lates  at  only  3  per  cent.  It  is  therefore  requisite  to  include  the 
present  value  of  the  investments,  viz.,  £993274  and  to  add 
thereto  the  sum  to  which  the  annual  income  will  accumulate 
at  the  end  of  the  period  at  the  amended  accumulation  rate.  As 
above  remarked,  it  is  not  necessary  to  consider  the  annual 
increment  in  connection  with  problems  involving  a  variation  in 
the  rate  of  income  from  investments  only,  but  later  in  Chapter 
XXYI,  when  dealing-  with  problems,  involving  a  variation  in 
the  period  of  repayment  complicated  by  a  variation  in  the 
accumulation  rate,  the  annual  increment  again  becomes  an 
important  factor.  The  annual  increment  has  been  considered 
in  this  exhaustive  manner  because  it  is  a  convenient  way  of 
expressing  the  resulting  correction  required  in  consequence  of 
any  of  the  above  variations. 

It  is  the  adjusted  annuity  under  the  amended  conditions 
which  is  the  equivalent  of  the  original  annuity  under  the 
previous  conditions.  It  may  be  divided,  at  both  periods,  into 
its  component  parts  of  :  — 

(1)  The  income  from  the  present  investments  received  from 

outside  sources,  and 

(2)  The  annual  instalment,  to  be  provided  out  of  revenue  or 

rate,  which  is  the  object  of  enquiry  in  all  cases. 

The  term  will  be  found  very  useful  when  dealing  with  all 
actual  adjustments,  since  by  dividing  the  accretions  to  the  fund, 
as  between  income  from  outside  investments  and  contributions 
from  internal  revenue,  a  clearer  insight  is  obtained  into  the 
principles  underlying  the  methods  adopted. 

Methods  of  Adjustment,  based  upon  the  Annual  Increment. 
(1)  The  Annual  Increment  {ratio)  Method.  It  will  be  gathered 
from  the  previous  remarks  that  an  adjustment  in  a  sinking  fund 
due  to  any  variation  in  the  original  conditions  may  be  made  in 
terms  of  the  annual  increment,  and  that  there  is  a  definite 
relation  always  existing  between  the  annual  increment  before 
adjustment  (the  present  annual  increment)  and  the  annual 
increment  after  the  necessary  adjustment  has  been  made  (the 
future  or  amended  annual  increment).  These  terms  are  fully 
defined  at  the  head  of  Chapter  X'XII,  where  the  component 
parts  of  each  annual  increment  are  exactly  described.  In  both 
cases  the  annual  instalment  may  be  found  by  deducting  from 
the  annual  increment  the  income  from  the  present  investments, 
thereby  eliminating  from  the  calculation  any  variation  in  the 


152  REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 

rate  per  cent,  of  income  from  investments,  and  confininjs^  tlie 

enquiry  to  tlie  variation  in  tlie  rate  of  accumulation  only.     Tlie 

annual  increment  may  be  considered  as  a  simple  annuity  to  be 

set  aside  for  a  number  of  years  (N)  and  accumulated  at  a  rate 

per  cent,  per  annum  expressed  by  the  factor  (R)  or  ratio,  and 

the  combination  of-tbese  factors,  as  regards  an  annuity  or  otber 

R^  —  1 
periodic  sum  is  expressed  by  tbe  formula    '-  tlie  derivation 

of  whicb,  from  the  simple  formula  A=P  R^,  is  fully  described 
in  Cbapter  VI.  There  is  an  exact  ratio  always  existing-  between 
a  given  annuity  to  be  accumulated  for  a  stated  number  of  years 
at  a  stated  rate  per  cent.,  and.  the  equivalent  annuity  to  be 
accumulated  for  a  varying  number  of  years,  at  a  varying  rate 
per  cent.,  depending  upon  tbe  respective  values  of  X  and  E,. 
This  is  the  basis  of  the  annual  increment  (ratio)  method, 
which  is  fully  described  in  Chapter  XXII,  and  which  has  been 
used  in  many  of  the  examples  in  the  following  chapters. 

(2)  The  Annual  Increment  {balance  of  loan)  Method.  In 
all  problems  involving  an  adjustment  in  a  sinking  fund  there 
are  two  fixed  factors  to  be  considered,  namely  :  — 

(1)  The  amount  of  loan  to  be  ultimately  repaid,  and 

(2)  The    amount   now    standing   to   the    credit    of    the    fund 

represented   by   the   present   investments. 

And  in  addition  there  are  two  variable  factors,  namely  :  — 

(1)  The  future  period  of  repayment  (N  years). 

(2)  The  future  rate  of  accumulation  of  the  fund  expressed  by 

the  factor  (R)  or  ratio. 

Any  variation  in  the  future  rate  of  income  to  be  received  upon 
the  present  investments  representing  the  fund  has  already  been 
eliminated  by  merging  such  annual  income  in  the  annual 
increment. 

In  all  problems  involving  a  variation  in  the  original 
conditions  governing  a  sinking  fund  the  subject  of  inquiry  is 
the  future  amended  annual  obligation,  and  this  may  be  ascer- 
tained by  reducing  the  present  factors  to  a  common  basis, 
namely,  the  balance  of  original  loan  which  will  be  unprovided 
if  the  amount  now  in  the  fund  be  immediately  applied  in 
redeeming  an  equivalent  part  of  the  loan  ultimately  repayable. 
The  balance  of  loan,  thereby  unprovided  for,  represents  the 
accumulated  amount  of  an  annuity  equal  to  the  future  or 
amended  annual  increment  to  be  set  aside  for  the  unexpired  or 
substituted  repayment  period  and  accumulated  at  the  original 


SINKING    FUND    PROBLEMS  153 

or  varied  rate  of  accumulation.  This  balance  of  loan  may  be 
ascertained  by  deducting  from  the  amount  of  loan  ultimately 
repayable  the  amount  now  in  the  fund  as  represented  by  the 
present  investments ;  and  the  future  annual  obligation,  which  is 
the  future  annual  increment,  may  be  ascertained  by  calculating, 
on  standard  form,  No.  3x,  the  sinking  fund  instalment  required 
to  provide  that  amount  under  the  altered  conditions,  both  as 
regards  the  period  of  repayment  and  the  rate  per  cent,  of 
accumulation.  The  amended  annual  increment  so  ascertained 
does  not,  however,  represent  the  amount  to  be  charged  annually 
against  the  revenue  or  rate  account  of  the  local  authority. 
The  conditions  governing  a  sinking  fund,  as  laid  down  in 
section  234  (5)  of  the  Public  Health  Act,  1875,  provide  that  if 
at  any  time  during  the  operation  of  a  sinking  fund  any  part  of 
such  fund  be  applied  in  redemption  of  debt,  the  local  authority 
shall,  out  of  its  annual  rate,  pay  into  the  sinking  fund  a  sum  at 
least  equal  to  the  amount  of  interest  Avhich  would  have  accrued 
to  the  fund  if  such  amount  had  not  been  so  applied.  Conse- 
quently the  future  amended  annual  instalment  is  found  by 
deducting  from  the  future  or  amended  annual  increment,  ascer- 
tained in  the  above  manner,  the  annual  income  to  be  received 
upon  the  present  investments  which  have  been  considered  as 
having  been  immediately  applied  in  the  redemption  of  an 
equivalent  part  of  the  loan,  whether  the  rate  of  income  upon 
such  investments  remains  unaltered  or  is  varied. 

This  is  the  basis  of  the  annual  increment  (balance  of  loan) 
method,  which  is  fully  described  in  Chapter  XXII,  and  which 
has  been  used  in  many  of  the  examples  in  the  following  chapters. 


154         REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 


CHAPTEE  XY. 

SINKING    FUND    PE(JBLE:\1S    RELATING    TO 
THE  AMOUNT  IN  THE  FUND. 

A  deficiency  in  the  fund  ;    how   it  may  arise  and  how  it 
may  be  adjusted. 

Peelimixaey  calculation  of  a  typical  sinking  fund  to  be 
used  to  illusteate  the  problems  to  be  discussed  in  the 
following  chapters.  methods  of  ascertaining  the 
position  of  a  sinking  fund  at  any  time.  a  deficiency  in 
the  fund  and  the  various  ways  in  which  it  may  be 
corrected.   general  summary  of  methods  of  adjustment. 

Before  considering-  in  detail  the  various  problems  arising  in 
connection  with  a  sinking  fund  it  should  be  stated  that  there 
are  in  each  case  several  methods  of  making  the  required  adjust- 
ment, all  of  which  depend  upon  the  present  position  of  the 
fund,  and  the  future  variation  in  the  original  conditions.  The 
subsequent  enquiry  will  include  variations  in  all  the  funda- 
mental factors  relating  to  such  a  fund,  namely,  the  amount  of 
the  fund,  the  period  of  repayment  of  the  loan,  the  rate  of 
accumulation  of  the  fund,  and  the  future  rate  of  income  to  be 
received  upon  the  present  investments  representing  the  fund. 
All  these  factors  have  each  their  own  effect  upon  the  ultimate 
function  of  the  fund,  namely,  the  repayment  of  the  loan,  but 
in  addition  they  act  and  react  one  upon  the  other. 

For  the  purpose  of  comparison,  therefore,  each  of  the  possible 
variations  will  be  considered  in  relation  to  one  and  the  same 
fund,  and  it  will  be  necessary  to  treat  all  the  problems  on,  as 
far  as  possible,  parallel  lines,  with  the  result  that  in  the  first 
instance  the  most  direct  method  of  making  the  adjustment  will 
not  be  discussed,  although  it  will  be  afterwards  fully  described. 
The  first  subject  of  enquiry  will  relate  to  the  simple  problem  of 
a  deficiency  in  the  amount  in  the  fund  without  any  further 
complication,  and  the  adjustment  of  such  a  deficiency  will  be 
made  by  the  deductive  method,  to  be  followed  later  when 
dealing  with  other  matters  affecting  the  fund. 

The  following  is  a  summary  of  the  general  rules  as  to  the 
adjustment  of  a  deficiency  in  a  sinking  fund  where  the  amount 
in  the  fund  only  is  in  question,  and  the  period  of  repayment, 


A    DEFICIENCY    IN    THE    FUND  I55 

the  future  rate  of  income  upon  the  present  investments,  and  the 
future  rate  of  accumulation  all  remain  unaltered.  In  this 
chapter  a  deficiency  in  the  fund  has  been  treated  in  a  very- 
exhaustive  manner,  perhaps  more  so  than  is  due  to  its  relative 
importance.  This  course  has  been  purposely  adopted  in  order 
to  demonstrate  the  practical  relation  between  the  various 
formulae  and  the  tables  deduced  therefrom. 


Summary  of  the  methods  of  adjustment. 

Variation  I  {^Deficiency),  in  which  the  adjustment  is  made 
by  an  additional  annual  instalment  to  he  set  aside  during 
the  whole  of  the  unexpired  portion  of  the  original  repayment 
period . 

Method  I.     The  deductive  method,  based  upon  all  the  factors 
governing  the  fund.  Statement  XV.  B. 

[1}  Calculate  the  amount  ichich  should  stand  to  the  credit  of 
the  fujid;  being  the  accumulation,  at  the  calculated  rate, 
of  the  annual  instalments  which  should  have  been  set 
aside.  Calculation  {XV)  2.    £9932-744. 

[2)  Ascertain  the  value  of  the  present  investments  representing 
the  fund,  including  in  the  case  of  a  local  authority,  the 
loan  repaid  by  means  of  the  sinking  fund.  £9463-00. 

{3}  The  difference  between  the  above  amounts  so  found  will 
be  the  deficiency  or  surplus  in  the  amount  of  the  fund  at 
the  time  of  making  the  enquiry.  £469-744. 

(4)  Calculate  the  amount  to  which  the  value  of  the  present 
investments  {as  in  2)  will  accumulate  at  the  end  of  the 
original  repayment  period. 

Calculation  {XV)  4.    £14799-71. 

{5)  Calculate  the  amount  of  the  remaining  original  annual 
sinking  fund  instalments  at  the  end  of  the  same  period. 

Calculation  {XV)  5.  £10960-62. 

{6)  Deduct  the  sum  of  the  two  amounts  so  obtained {£25760-33) 
from  the  amount  of  the  original    loan. 

(7)  The  difference  represents  the  amount  of  loan  which  will 
be  unprovided  for  in  the  case  of  a  deficiency,  or  provided 
for  in  excess,  in  the  case  of  a  surplus,  at  the  end  of  the 
original  repayment  period  {actually  £7346o9).    £734-67. 


156  REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 

[8)  Calculate  the  additional  annual  sinking  fund  instalment 

required  to  inovide  this  sum  at  the  end  of  tJic  jcpayme7it 
period.  Calculations  {XV)  3  and  {XVI)  1.    £4o-594. 

(9)  Adjust  the  original  sinJcing  fund  instalment  hy  adding  to 

it  the  annual  instalment  so  obtained  in  (8)  in  the  case  of 
a  deficiency  or  by  deducting  it  in  the  case  of  a  surplus. 

[10)  Prepare  a  statement  sJwiving  the  final  repayment  of  the 
loan  by  the  operation  of  the  sinJdng  fund  under  the 
amended  conditions.  Statement  -YT'7.  A. 

{11)  Prepare  a  pro  forma  account  shoiving  the  amount  icliich 
should  be  in  the  fund  at  the  end  of  each  year  of  the 
unexpired  portion  of  the  repayment  period  for  future 
reference.  Pro  forma  Account,  To.  2,  Chapter  XVI . 

Method  II.     In   which  the  original  instalment  does  not  enter 
into  the  calculation.  Statement  XVI.  A. 

{1)  Calculate  the  amount  to  which  the  sum  which  should  be 
in  the  fund,  as  found  by  Calculation  {XV)  2,  will 
accumulate  at  the  end  of  the  repayment  period. 

Calculation  (XVII)  2.     £loo34-37o. 

{2)  Calculate,  and  deduct  from  the  sum  so  found,  the  amount 
to  lohich  the  value  of  the  present  investments  (£9,463) 
will  accumulate  at  the  end  of  the  repayment  period. 

Calculation  {XV)  4.    £14799-71. 

{3)  The  difference  will  be  the  amount  of  original  loan  which 
will  be  unprovided  for  in  the  case  of  a  deficiency  or 
provided  for  in  excess  in  the  case  of  a  surplus  {as  found 
in  No.  7,  Method  1).    {actually  £7 34' 669.)         £734  66-5. 

{4)  Adjust  the  or-iginal  instalment,  as  in  Xos.  8  and  9  in 
Method  1,  above. 

{5)  Prepare  a  statement  showing  the  final  repayment  of  the 
loan  by  the  operation  of  tJte  sinJxing  fund  under  the 
amended  conditions.  Statement  XVI.  A. 

{6)  Prepare  a  pro  forma  account,  as  mentioned  above. 

Xo.  2,  Chapter  XVI. 

Method  III.         The  dieect  method,   based  entirely   upon   the 
present  position  of  the  fund.  Statement  X^  I.  A. 

{1)  Calculate  the  amount  ivhich  sliould  stand  to  the  credit  of 
the  fund,  being  the  accumulation  at  the  calculated  rate, 
of  the  annual  instalments  ulilch  should  have  been  set 
aside.  Calculation  {XV)  2.    £9932-74. 


A    DEFICIENCY    IN    THE    FUND  157 

(2)  Ascertain  the  value  of  the  -present  investments  represent' 

the  fund,  including,  in  the  case  of  a  local  authority  the 
loan  repaid  by  means  of  the  sinking  fund.  £946S-00. 

(3)  The  difference  between  the  above  amounts  so  found,  will 

be  the  deficiency  or  surplus  in  the  amount  of  the  fund  at 
the  time  of  making  the  enquiry.       Deficiency.  £469-744. 

(4)  Calculate  the  annuity  or  annual  instalment  of  which  this 

sum  is  the  p^'csent  value,  depending  upon  the  period  over 
which  the  correction  shall  extend. 

Calculation  {XV)  3.     £45-594. 

(J)  Adjust  the  original  sinJying  fund  instalment  by  adding 
to  it  the  instalment  so  obtained  in  the  case  of  a  deficiency 
or  by  deducting  it  in  the  case  of  a  surplus. 

(6)  Prepare  a  statement  showing  the  final  repayment  of  the 

loan  by  the  operation  of  the  fund  under  the  amended 
conditions.  Statement  XVI.  A. 

(7)  Prepare  a  pro  forma  account,  as  mentioned  above. 

^  ^  No.  2,  Chapter  XVI. 


Method  IV.  The  anxial  increment  (balance  or  loan)  method, 

based  upon  the  future  annual  increment  and  the  present 

position  of  th e  fund .  Statement  XV I. ^  B. 

This  method  will  be  fully  discussed  in  Chapters  XVI 

and  XXII . 

{!)  Ascertain  the  value  of  the  present  investments  represent- 
ing the  fund,  including  in  the  case  of  a  local  authority, 
the  loan  repaid  by  means  of  the  fund,  as  already 
described.  £9463-00. 

{2)  Deduct  the  value  so  obtained  from  the  amount  of  original 
loan  repayable  at  the  end  of  the  prescribed  period. 

£2649500. 

(3)  The  remainder  represents  the  balance  of  original  loan  to 
be  provided  by  the  accumulation  of  the  future  or  amended 
annual  increment,  xi/uo^  uu. 

{4)  Calculate  the  annuity,  or  annual  increment  required  to 
provide  the  remainder  so  found,  at  the  end  of  the  pre- 
scribed period  at  the  future  rate  of  accumulation. 

Calculation  [XVI)  9.     £1057-033. 


158    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

(<5)  Deduct  therefrom  the  future  annual  income  to  he  received 
from  the  i}resent  investments.  £331' 205. 

(6)  'The  remainder  will  he  the  am,ended  annual  instalment  to 

he  provided  out  of  revenue  or  rate  instead  of  the  original 
instalment.  £725'828. 

The  difference  hetiveen  the  two  instalments  will  he  the 
additional  annual  instalment  found  hy  either  of  the  fjre- 
ceding  methods.  £4o'594 

(7)  Prepare  a  statement  showing  the  final  repayment  of  the 

loan  hy  the  operation  of  the  sinking  fund  under  the 
amended  conditions.  Statement   XVI.   A. 

(5)  Prepare  a  pro  forma  account,  as  mentioned  ahove. 

No.  2,  Chapter  XVl. 

Note,  Calculations  {XV)  1  and  {XV)  2  are  given  in  full 
at  the  end  of  this  Chapter.  The  remainder  are  given,  in  an 
ahhreviated.  form,  in  the  Appendix. 


Calculation  of  a  Typical  Sinking  Fund.  The  previous 
chapter  contains  a  brief  summary  of  the  nature  of  the  problems 
likely  to  arise  with  regard  to  sinking  funds  of  all  kinds  both  in 
connection  with  local  authorities  and  commercial  or  financial 
undertakings.  There  may  be  at  times  a  combination  of  the 
several  variations,  but  in  the  first  instance  each  problem  will 
be  considered  alone,  deferring  the  examination  of  more  compli- 
cated cases.  In  order  to  do  this  in  a  consecutive  manner,  an 
imaginary  sinking  fund  Mill  be  adopted  which  will  be  used  to 
illustrate  the  whole  of  the  examples  to  be  afterwards  considered, 
because  by  this  means  only  is  it  possible  to  apply  the  results 
obtained  in  considering  the  simpler  problems,  to  those  of  a  more 
complex  nature.  It  will  be  assumed  that  the  sinking  fund  is  in 
respect  of  a  loan  of  £26,495,  payable  at  the  end  of  a  period  of 
25  years,  and  that  the  instalments  will  be  set  aside  annually, 
and  will  accumulate  by  investment  at  3|  per  cent,  per  annum. 
The  first  step  is  to  ascertain  the  annual  instalment,  and  the 
calculation  will  bo  made  upon  standard  calculation  form  No.  3x, 
by  the  three  methods  described  at  the  head  of  Cliapter  XIII. 
See  Calculation  (XV)  1.  In  this  case,  as  in  all  others,  the 
method  by  formula  is  shown  because  although  the  methods  by 
table,  including  Thoman's,  are  much  shorter,  yet  all  the 
published  tables  contain  only  a  limited  number  of  rates  percent. 


A    DEFICIENCY    IN    THE    FUND  159 

Furtlier,    the   tables   are   not    of   mucli    assistance   when    it    is 
necessary  to  ascertain  the  rate  per  cent,  or  the  number  of  years 
with   accuracy,   which   can   only   be   done   by  the   method   by 
formula,    and   then   sometimes   only   approximately.      Anyone 
depending  upon  the  published  tables  alone  without  a  knowledge 
of  the  method  by  formula  is  at  a  great  disadvantage  when  the 
book  of  published  tables  is  absent.     An  acquaintance  with  the 
methods  by  formula  and  a  table  of  logs,  is  all  that  is  required, 
and  a  very  small  memorandum  book  will  contain  the  whole  of 
the  f ormulse  mentioned  in  this  work,  which  will  be  found  at  the 
head  of  the  various  chapters,  and  also  in  Chapter  X  dealing 
with  the  standard  calculation  forms  prepared  by  the  author. 
There  is  a  further  advantage  gained  by   a  knowledge  of  the 
formula  and  how  they  are  arrived  at,  namely,  a  clear  under- 
standing of  the  principles  underlying  the  theory  of  compound 
interest  which  renders  it  an  easy  matter  to  make  all  calculations 
by   one   or  more   alternative   methods   and   thereby   prove   the 
accuracy  of  the  results   obtained.      In   making   a   calculation 
similar  to  the  foregoing  in  which  it  is  necessary  to  multiply  or 
divide  a  large  principal  sum  by  a  figure  containing  5  places  of 
decimals  it  is  important  to  be  extremely  careful  to  obtain  the 
exact  logs,  or  antilogs.  by  means  of  the  tables  of  proportional 
parts  which  will  be  found  in  the  margin  of  the  log.   tables. 
In  the  above  instance,  and  in  all  other  cases  where  it  is  required 
to  find  the  log  of  U^ ,  the  log.  of  R  should  be  carefully  ascer- 
tained, especially  as  to  the  last  3  or  4  figures.     In  order  to 
obtain  the  Nth  power  of  R,  the  log.  of  E  is  multiplied  by  25, 
and  any  error  in  the  last  two  figures  will  have  a  material  effect 
upon  tiie  result  so  found  by  multiplication.     For  this  reason, 
in  Table  Y.  (A.),   in  Chapter  Y,   containing  the  values  of  (R) 
for  various  rates  per  cent,  the  corresponding  logs,  of  (E)  are 
given  to  eight  places  instead  of  seven  as  in  the  usual  log.  tables. 
These  logs,  may  be  multiplied  by  the  number  of  years  and  the 
seventh  figure  adjusted,   leaving  out  the  eighth  figure.      The 
logs,  of  RN  are  given  in  Thonian's  tables  for  many  rates  per 
cent.,  and  even  in  cases  where  the  method  by  formula  is  used  it 
may  be  taken  direct  from  Thoman's  tables  with  a  saving  of 
time.     The  logs,  of  (RN-1)  cannot  be  found  from  the  tables, 
but  only  by  calculation,  although  the  actual  values  of  R^-l 
may  be  found  by  deducting  unity  from  the  actual  values  given 
in  Table  I. 

Calculation  (XY)  1  shows  that  an  annual  instalment  of 
£680-234  is  required,  and  the  pro  forma  account  No.  1  at  the 
end  of  this  chapter  shows  the  normal  accumulation  of  the  fund. 


i6o         REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

Method  of  Asceetaixixg  the  Position  of  a  Sinking  Fund 
AT  ANY  Time.     Having  ascertained  tliat  an  annual  sinking  fund 
instalment  of  £680"234  is  required  to  be  set  aside  and  accumu- 
lated at  3i  per  cent,  per  annum  for  25  years  to  repay  a  loan  of 
£26,495,  at  the  end  of  tliat  period,  this  information  will  now  be 
applied  to  an  enquiry  into  the  position  of  the  fund  at  the  end 
of  the  12th  year.     In  an  investigation  of  this  nature  occurring 
in  actual  practice  the  annual  instalment  would  of  course  be  the 
basis  of  the  enquiry  as  it  would  have  been  in  operation  for  a 
period  of  12  years.     The  first  stage  of  the  actual  enquiry  is  to 
ascertain  the  amount  which  should  now  stand  to  the  credit  of 
the  fund,   on  the  assumption  that  the  annual  instalment  has 
been  regularly  set  aside  and  has  been  promptly  invested  at  the 
end  of  each  year,  to  yield  3|  per  cent.     This  amount,  as  shown 
by  Calculation  (XY)  2,  should  be  £9932-744.     The  next  step 
is  to  ascertain  the  actual  amount  standing  to  the  credit  of  the 
fund  in  the  books  of  the  local  authority  or  private  undertaking 
and  then  to  compare  this  amount  with  the  actual  value  of  the 
investments  representing  the  fund,  including  in  the  case  of  a 
local  authority  the  loans  redeemed  by  means  of  the  fund.     In 
the  case  of  a  commercial  or  financial  undertaking  there  may  not 
be  any  obligation  to  invest  the  fund  in  specific  outside  securities, 
and  the  amounts  to  be  charged  annually  against  the  profits  of 
the  concern  may  be  allowed  to  remain  uninvested  and  go  to 
swell  either  the  floating  or  fixed  assets.     In  such  a  case  it  may, 
and  will  most  probably,  happen  that  the  book-keeping  has  been 
correct,  and  that  the  profit  and  loss  account  of  the  undertaking 
has  been  each  year  charged  with  the  proper  annual  instalment 
and  also  with  the  proper  annual  interest  upon  the  increasing 
balances  to  the   credit   of   the   fund.     Under   such   conditions 
there  will  rarely  be  any  necessity  for  enquiry  seeing  that  the 
fund  will  always  stand  in  the  books  at  the  correct  amount,  and 
any  deficiency    of    assets    representing   the    fund    will    not    be 
apparent,  but  will  be  merged  in  the  general  state  of  the  assets 
of  the  concern.     But  in  the  case  of  commercial  and  financial 
undertakings,  where  there  is  an  obligation  to  take  the  amount 
of   the   annual   instalments   out   of   the   floating   assets   of   the 
concern  and  invest  the  same  in  specific  outside  securities,  the 
case  is  exactly  similar  to  the  conditions  imposed  by  Parliament 
upon   all  local   authorities,   and   may   be   treated   on   precisely 
similar  lines.     The  deficiency  in  both  cases  may  arise  in  twO' 
ways,  even  if  the  annual  instalments  have  been  regularly  set 
aside  and  the  proper  amount  of  money  actually  paid  into  the 
sinking  fund  account.     The  first  cause  of  the  deficiency  may  be 


A    DEFICIENCY    IN    THE    FUND  i6i 

tiiat  owing-  to  delay  in  investing  the  instalments,  or  owing  to  a 
fall  in  tlie  rate  of  income  received  from  the  investments,  the 
fund  has  not  accumulated  at  the  rate  originally  anticipated  and 
upon  which  the  calculation  of  the  original  annual  instalment 
was  based.  The  second  cause  of  the  deficiency  may  he  that  the 
investments  have  depreciated  in  value  and  cannot  now  be 
considered  as  representing  the  amount  standing  to  the  credit  of 
the  fund,  and  there  may  have  been  in  addition  an  actual  loss 
on  realisation.  But  it  is  necessary  to  go  further  and  ascertain 
whether  these  investments  will,  or  will  not,  as  far  as  can  be 
judged,  be  of  such  a  value  at  the  end  of  the  repayment  period 
that  they  will  fulfil  the  original  purpose  of  redeeming  their 
proportion  of  the  loan.  As  already  remarked  in  dealing  with 
the  present  investments  in  Chapter  XIY,  this  is  a  very  difficult 
matter  if  the  unexpired  repayment  period  is  a  long  one ;  and  it 
is  therefore  the  general  practice  to  assume  the  future  estimated 
rate  of  accumulation  on  the  low  side,  leaving  any  further 
adjustment  to  be  made  at  a  later  date  when  the  conditions  will 
be  better  known.  In  the  case  of  local  authorities,  as  will  be 
seen  by  a  perusal  of  Article  11  (2)  of  the  County  Stock  Eegula- 
tions  of  1891,  the  Local  Government  Board  are  empowered  to 
take  cognisance  of  such  matters,  and  the  same  supervision  may 
be  said  to  apply  to  the  whole  of  the  loans  of  local  authorities. 
In  the  case  of  commercial  or  financial  undertakings  the 
adequacy  or  otherwise  of  these  investments  and  of  the  fund 
generally  would  be  investigated  by  the  auditors  of  the  company 
or  by  or  on  behalf  of  the  loan  holders.  In  the  present  chapter 
it  will  be  assumed  that  there  is  a  deficiency  in  the  sinking  fund 
of  a  definite  amount  arising  from  any  of  the  above  causes,  but 
for  the  present  the  problem  will  not  be  complicated  in  any  way 
by  a  variation  in  the  period  of  repayment  or  in  the  future  rates 
per  cent,  of  income  or  of  accumulation. 

The  Yaeious  Methods  of  Correcting  a  Deficiency  in  a 
Sinking  Fund.  Having  assumed  that  there  is  now  an  actual 
ascertained  deficiency  in  the  sinking  fund  the  various  methods 
will  now  be  considered  by  Avhich  it  may  be  made  good.  In  the 
case  of  a  local  authority  such  a  deficiency  may  often  arise,  but 
generally  it  is  of  small  amount  due  entirely  to  a  reduction  in 
the  rate  of  income  on  part  of  the  fund  uninvested  and  in  the 
bank.  In  practice  this  is  met  by  charging  any  such  deficiency 
to  the  general  revenite  or  rate  account  of  each  year.  If  the 
deficiency  in  the  case  of  a  local  authority  is  large,  owing  either 
to  serious  omissions  in  previous  years  or  to  the  accumulation  of 


i62    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

many  small  annual  deficiencies,  the  matter  wonld  be  decided  by 
tlie  Local  Government  Board  or  by  Parliament  when  next 
powers  are  soug'ht  by  special  Act.  This  need  not  now  be 
discussed  in  detail  because  all  the  available  methods  will  be 
fully  described  later.  Taking  actual  fig'ures,  it  will  be  assumed 
that  the  above  imaginary  sinking  fund  (requiring  an  annual 
•nstalment  of  £680-234  to  repay  £26,495  in  25  years  at  an 
accumulation   rate    of   3|    per   cent.)    amounts    at   the    end   of 

12  years  to      £9463000 

instead  of  the  correct  amount  shown  by  Calculation 

XY.  (2)    '    £9932-744 


or  a  deficiency  of       ...     £469-744 


and  that  the  conditions  governing-  the  fund  require  that  this 
deficiency  should  be  made  good  in  some  manner  out  of  rate  or 
revenue,  or  out  of  profits  in  the  case  of  a  commercial  or  financial 
undertaking. 

There  are  several  ways  in  wliich  such  a  deficiency  may  be 
corrected,  namely:  — 

(a)  By  an  immediate  payment  of  the  deficiency  of  £469'744 
into  the  fund,  which  need  not,  however,  be  considered, 
because,  although  the  soundest  financially,  it  has  no 
bearing  upon  the  subject  under  review. 
(6)  By  an  additional  annual  sinking  fund  instalment  to  be 
spread  over  the  whole  of  the  unexpired  13  years  of  the 
original  repayment  period,  in  augmentation  of  the 
original  annual  instalment  of  £680-234. 

(Variation  I,  Chaptei  XYI.) 

(r)   By  an  additional  annual  sinking  fund  instalment  to  be 

spread  over  a  shorter  period  than  the  full  unexpired  term 

of  13  years.  (Variation  II,  Chapter  XVI.) 

Having  dismissed  the  correction  by  an  immediate  payment 

into  the  fund,  the  last  two  alternatives  will  be  applied  to  the 

imaginary  deficiency  in  order  to  ascertain  the  corrected  annual 

instalment  consequent  thereon.    The  above  deficiency  of  £469-744 

represents  an  amount  of  money  ])ayable  now,  being  the  amount 

(in  the  sense  in  Avhich  it  is  used  in  Table  III)  of  past  annual 

omissions  accumulated  at  3|  per  cent.     It  does  not  represent  an 

equivalent   amount    of   the   original    loan,    as    shown    later   by 

Calculation  (XV)  6.     Stated  in  terms  of  the  original  loan,  it  is 

the  present  value  at  3^  per  cent,  per  annum  of  £734*659,  part 

of  that  loan,  repayable  in  13  years  from  the  present  time,  the 


A    DEFICIENCY    IN    THE    FUND  163 

repayment  of  winch  lias  not  in  the  past  been  provided  for  as  it 
should  have  been. 

The  several  methods  of  adjusting  the  deficiency  given  in 
summary  form  at  the  head  of  this  chapter  will  now  be 
described  in  detail,  commencing  with  the  direct  method,  III, 
which  is  the  simplest,  after  which  Method  I  will  be  considered, 
followed  by  Method  II,  leaving  Method  lY  to  be  dealt  with  in 
the  following  chapter. 

Method  III,  The  ^^resent  deficiency  of  £469" 744,  if  not 
complicated  by  other  varying  factors  of  time  or  rate  per  cent., 
may  be  regarded  in  its  simplest  form  as  the  present  value  of  an 
additional  future  annual  instalment  required  to  be  set  aside  and 
accumulated  during  the  unexpired  portion  of  the  original  re- 
payment period  in  augmentation  of  the  original  instalment ; 
and  in  the  summary  of  methods  at  the  head  of  this  chapter  this 
is  described  as  the  direct  method  No,  III.  The  additional  annual 
instalment  is  found  by  Calculation  (X.\)  3,  which  shows  that  the 
deficiency  of  £469 "744  is  the  present  value  of  an  additional 
annual  instalment  of  £45'594  to  be  set  aside  and  accumulated 
at  3|  per  cent,  during  the  unexpired  13  years  of  the  original 
repayment  period.  The  same  result  is  obtained  by  Calculation 
(XYI)  1  in  the  following  chapter,  which  shows  that  the  annual 
instalment  which  will  amount  to  £734' 659  of  original  loan  at 
the  end  of  the  period  is  also  £45"594.  The  above  amount 
(£734'659)  of  original  loan  (by  Calculation  (XY)  6  in  this 
chapter)  is  shown  to  be  the  accumulated  amount  of  the  present 
deficiency  of  £469' 744. 

Method  I.  The  investigation  will  now  be  continued  on  the 
lines  set  out  in  Method  I,  at  the  head  of  this  chapter.  The 
present  position  of  the  fund  may  be  stated  in  terms  of  the 
present  value  of  each  of  the  component  parts  of  the  fund, 
namely,  the  present  investments,  the  deficiency,  and  the 
remaining  original  annual  instalments.  Seeing,  however,  that 
the  object  of  the  fund  is  to  repay  the  loan,  and  that  other  causes 
of  adjustment  all  affect  the  ultimate  amount  of  the  loan,  the 
effect  will  be  more  clearly  shown  by  reducing  the  whole  of  the 
factors  in  all  cases  to  terms  of  loan,  repayable  at  the  end  of  the 
prescribed  period.  This  will  require  three  calculations,  as 
follows: — (1)  Ascertain  the  sum  to  which  the  present  invest- 
ments (£9,463)  will  accumulate  at  the  end  of  the  unexpired 
period  of  13  years  at  3^  per  cent.  See  Calculation  (XY)  4. 
(2)  Add  to  this  amount  the  sum  to  which  the  remaining  original 


i64    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

annual  instalments  of  £680'234  will  amount  at  the  end  of  the 
same  period,  also  accumulated  at  3|  per  cent.     See  Calculation 

(XV)  5.  The  sum  of  these  two  factors  will  represent  the 
reduced  portion  only  of  original  loan  which  would  be  provided 
if  the  present  deficiency  were  not  corrected.  This  total  added 
to  the  amount  of  £734'659  to  which  the  present  deficiency  of 
£469'744  would  accumulate  in  13  years  at  3^  per  cent,  [see 
Calculation  (XY)  6]  will  make  up  the  total  amount  of  the 
original  loan.  This  last  factor  is  the  measure  of  the  deficiency 
expressed  in  terms  of  original  loan,  and  may  be  treated  in  the 
same  way  as  the  full  amount  of  the  loan,  in  Calculation  (XV)  1, 
to  find  the  original  annual  instalment.  The  required  annual 
instalment  so  found,  namely,  £45'594,  represents  the  additional 
annual  sum  to  be  set  aside  and  accumulated  in  augmentation 
of  the  original  annual  instalment  of  £680'234.     See  Calculation 

(XVI)  1. 

The  three  calculations  to  show  the  equivalent  amounts  of 
original  loan  will  be  made  as  before  by  formula  and  logs.,  and 
also  by  Table  III  and  Thoman's  tables.  There  is  really  not 
any  necessity  to  prove  the  result  by  further  calculation  because 
the  above  results,  added  together,  should  be  equal  to  the  total 
amount  of  original  loan  to  be  provided  at  the  end  of  the 
prescribed  period. 

Method  II.  As  already  stated  in  the  summary  at  the  head 
of  this  chapter,  the  sum  of  £T34"659,  being  the  amount  of  loan 
which  will  remain  unprovided  if  the  present  deficiency  be  not 
corrected,  may  also  be  ascertained  by  leaving  out  of  account  the 
future  original  annual  instalments  (which,  yer  sc,  are  unaffected 
by  any  present  deficiency  in  the  fund),  and  comparing  the 
ultimate  accumulated  amount,  at  the  end  of  the  period,  of  the 
present  investments  of  £9463' 00  with  the  accumulated  amount 
of    the    sum    of    £9932'744    which,    as    shown    by    Calculation 

(XVII)  2,  should  have  been  in  the  fund  if  the  original  anticipa- 
tions had  been  realised,  as  follows:  — 

The  ultimate  amount  of  the  present  iuA-estments  of 
<£9463-00,  as  shown  by  Calculation  (XV)  4, 
will  be £14799-710 

and,  in  Calculation  (XVII)  2,  it  is  shown  that  the 
above  sum  of  £9932' 744  will  in  13  years  at 
^  per  cent,  amount  to     £15534-375 


a  difference  of  (actually  £734-659)         £734-665 


A    DEFICIENCY    IN    THE    FUND  165 

wliicli,  as  proved  by  Calculation  (XY)  6,  is  the  iiltiiuate  amovmt 
of  loan  represented  by  the  present  deficiency  of  £469" 744.  The 
following  summary  Avill  make  the  matter  clear:  — 


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i66    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Calculation  (XV)  3  sliows  tliat  the  above  deficiency  of 
£469' 744  is  the  equivalent  present  value  of  an  annual  instalment 
of  £45o94,  which,  accumulated  for  13  years  at  3^  per  cent., 
will,  as  shown  by  Calculation  (XYI)  1  in  the  following  chapter, 
provide  the  above  portion  namely  £734" 659,  of  the  original  loan. 

The  following  Statement  XY.  B.  shows  the  present  position 
of  the  fund,  and  also  the  amount  of  loan  which  will  be  provided 
at  the  end  of  the  unexpired  portion  of  the  repayment  period, 
namely,  13  years,  by  the  accumulation  of  the  amount  now 
standing  to  the  credit  of  the  fund  to  be  increased  by  the 
remaining  original  annual  instalments,  but  without  any 
correction  being  made  to  adjust  the  present  deficiency  of 
£469"  744.  The  final  repayment  of  the  loan  after  correcting  the 
present  deficiency  by  an  additional  annual  instalment  will  be 
shown  in  Statement  XYI.  A.  in  the  following  chapter. 


A    DEFICIENCY    IN    THE    FUND  167 

A  Deficiency  in  the  Fund,  Statement  XV.  B. 

The   Deductive  Method.     No.  i. 

Showing  the  position  of  the  fund  at  the  end  of  the  12th  year, 
and  the  amount  of  loan  which  will  be  unprovided  at  the 
end  of  the  repayment  period  if  the  present  deficiency  be 
allowed  to  accumulate,  instead  of  being  immediately 
corrected  by  an  additional  annual  instalment. 

Present  investments  (at  end  of  12th  year)  £946300 

•   Amount  thereof,  accumulated  for  13  years  at 

3i  per  cent.  Calculation  (XV)  4  £14799-71 

Original  annual  instalment  : — 

Amount  of  £680'234  per  annum,  for  13  years  at 

3i  per  cent.  Calculation  (XV)  5  £10960-62 


Provision  already  made  will  repay  loan  of  ...     ...  £25760-33 

Deficiency,  being  the  balance  of  loan  unprovided 
for,  represented  by  the  present  deficiency  of 
£469-744,  accumulated  for  13  years  at  3^  per 
cent,   (actually  £734-659)    Calculation  (XV)  6  734-67 


Amount  of  original  loan       £2649500 


Additional  annual  instalment  required, 

Calculations  (XV)  3  and  (XVI)  1         £45-594 


Amended  annual  instalment, 

Original  annual  instalment       £680-234 

Additional  annual  instalment 45-594 


£725-828 


The  final  repayment  of  the  loan  by  the  operation  of  the 
sinking  fund  after  making  the  above  adjustment  in  the  annual 
instalment  is  shown  in  Statement  XVI.  A.,  and  by  the  pro 
forma  account,  No.  2,  Chapter  XVI. 


i68 


REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 


Pro  forma  Sinking  Fund  Account,  No.  1. 

Loan  of  £26,495,  re  payable  at  the  end  of  2-5  years. 

Annual    Instalment.         Calculation    (XY)  1.         £680-234. 
Eate  of  Accumulation,  3|  per  cent. 

Sliowino;  the  normal  accumulation  of  tlie  fund. 


Year. 

Amount  in 

tlie  fund 

at  beginning 

of  year. 

Income 

received  from 

investments 

3^-  per  cen£. 

.\nnual 
sinkiaig  fund 
Instalment. 

A)i;ount  in 
the  fund 

at 
end  of  year. 

Year 

1 

Nil 

Nil 

680-234 

680-234 

1 

2 

680-234 

23-808 

680-234 

1384-276 

•  2 

3 

1384-276 

48-450 

680-234 

2112-960 

3 

4 

2112-960 

73-954 

680-234 

2867-148 

4 

5 

2867-148 

100-350 

680-234 

3647-732 

5 

6 

3647-732 

127-671 

680-234 

4455-637 

6 

4455-637 

155-947 

680-234 

5291-818 

7 

8 

5291-818 

185-213 

680-234 

6157-265 

8 

9 

6157-265 

215-504 

680-234 

7053-003 

9 

10 

7053-003 

246-853 

680-234 

7980-090 

10 

11 

7980-090 

279-302 

680-234 

8939-626 

11 

12 

8939-626 

312-884 

680-234 

9932-744 

12 

13 

9932-744 

347-648 

680-234 

10960-626 

13 

14 

10960-626 

383-622 

680-234 

12024-482 

14 

15 

12024-482 

420-857 

680-234 

13125-573 

15 

16 

13125-573 

459-395 

680-234 

14265-202 

16 

17 

14265-202 

499-282 

680-234 

15444-720 

17 

18 

15444-720 

540-565 

680-234 

16665-519 

18 

19 

16665-519 

583-293 

680-234 

17929-046 

19 

20 

17929-046 

627-517 

680-234 

19236-797 

20 

21 

19236-797 

673-268 

680-234 

20590-299 

21 

22 

20590-299 

720-661 

680-234 

21991-194 

22 

23 

21991-194 

769-692 

620-234 

23441-120 

23 

24 

23441-120 

820-439 

680-2:54 

24941-793 

24 

25 

24941-793 

872-973 

680-234 

26495000 

25 

A    DEFICIENCY    IN    THE    FUND 


169 


Calculation  (XV)  1. 

Standard  Calculation  Form,  No.  3x. 

To  find  the  annual  sinking  fund  instalment  to  be  provided  out 
of  revenue  or  rate  to  repay  the  loan  under  the  original 
conditions  laid  down  at  the  time  of  borrowing. 

Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
as  a  sinking  fund  at  3^  per  cent,  per  annum  to  provide 
£26,495  in  25  years. _____^ 

(A)     By  Formula.          Ky  =  ^i  (^  ^p^-^^Rule  1,  Chapter  XIII. 
Lost.  Ratio 


Log. 
RN-1 


Multiply  Log.  R  by 


Convert  Log. 

to  ordinary  number 
deduct  unity 

Log.  of  this  is 

Log.  Amount  of  Loan 
add  Log.  r 


deduct  Log.  (R^  - 1)  above 


R 

N 

1-035 
25 

0-0149403 
25 

RN 

(l-035)-^5 

0-3735087 

RN 

-1 

2-36324 
1- 

RN-1 

1-36324 

0-1345738 

M 

T 

26,495 
0035 

4-4231639 
2-5440680 

Mr 

RN-1 

2-9672319 
0-1345738 

A.V 

2-8326581 

Required  annual  ii)stalment,  £680-2336 


M 


(B)     By  Table  III.         Ai/-  RN  -  1  Rule  2,  Chapter  XIII. 


Log .  Amount  of  Loan 
Table  III.  25  years,  3^  per  cent. 
Amount  of  £1  per  annum 
ded^ict  Log. 


M                26,495 

RN-1     38-94986 
r 

4-4231639 
1-5905058 

Ay 

2-8326581 

Required  annual  instalment,  £680-2336 


(C)     By  Thoman's  Table.     Ay  =  M  (^  \ Rule  3,  Chapter  XIII. 
3^  per  cent.,  25  years.  ^^ 


Log.  Amount  of  Loan 
add  Loff.  a^ 


deduct  Log.  RN  in 
Table +  10 


M 


26,495 


4-4231639 
8-7830029 


M  a" 

RN 


13-2061668 
10-3735087 


A.v 


2-8326581 


Required  annual  instalment,  £680-2336 


lyo    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Calculation  (XV)  2. 

Standard  Calculation  Form,  -A  o.  3. 

To  find  the  amount  whicli  sliould  stand  to  the  credit  of  a 
sinking-  fund   at   any  time. 

Required  the  amount  which  shoukl  stand  to  the  credit  of  a 
sinking-  fund  representing-  the  accumulation  of  an  annual 
instalment   of   £680"2o4   for   12  years   at   3^   per  cent. 

(A)     By  Formula.          M  =  A2/(^^~^)     Rule  1,  Chapter  YI. 


Log 

RN-1 


Log.  Ratio 

Ji/uUiph/  Log.  R  by 


Convert  Log. 

to  ordinary  number 
deduct  unity 

Log.  of  this  is 

Log.  Annuity 
add  Loff.  R^  —  1  above 


deduct  Log,  r 


1035 
12 


00149403 
12 


RN 


(1-035)12        0-1792842 


RN 

-1 


1-51107 
1- 


RN-1       0-51107 


1-7084792 


RN-1 


680-234 


^-8326581 
1-7084792 


AyCR^-1] 


0-035 


2^-5411373 
2-5440680 


M 


3-9970693 


Required   amount,    £9932- 744. 


(B.l 


By  Table  III.         M  =  Ay  (  — ^_  ^  )    Rule  2,  Chapter  YI. 


Table  III.    12  years,  3|  per  cent. 
Amount  of  £1  per  annum 
add  Log.  Annuity 


RN-1     14-60196 
A^  680-234 


1-1644112 
2-8326581 


M 


3-9970693 


Required   amount,    £9932-744. 


(Cj     By  Thoman's  Table.      M  =  A?/('^  ^      Rule  3,  Chapter  YI. 
3i  per  cent.,  12  years. 

Log.  Annuity 
add  Log.  RN  in 

Table +10 

deduct  Log".  «" 


Ay 

RN 

680 

234 

2-8326581 
10-1792842 

AyRN 

130119423 
9-0148730 

M 

3-9970693 

Required    amount,   £9932-744. 


THE    CORRECTION    OF    A    DEFICIENCY  171 


CHAPTEE  XYI. 

SINKING    FUND    PROBLEMS    RELATING    TO 
THE  AMOUNT  IN  THE  FUND. 

The  correction  of  a  deficiency  in  the  fund. 

Variation  I. 

By  an  additional  annual  instalment  to  be  set  aside 
during  the  whole  of  the  unexpired  portion  of  the 
repayment  period.  statement  xyi.  a. 

Variation  II. 

By    an    ADDITIONAL    ANNUAL    INSTALMENT    TO    BE    SET    ASIDE 

during  the  earlier  part  only  of  the  unexpired  portion 
of  the  repayment  period.  statement  xvi.  c. 

Summary  of  the  methods  of  adjusting  a  deficiency.     The 
several   methods   described.        tlie   annual   increment 

(balance    of    loan)     METHOD.  STATEMENT     SHOWING     THE 

FINAL   REPAYMENT    OF    THE    LOAN  BY    THE    OPERATION    OF    THE 

AMENDED  ANNUAL  INSTALMENT,  IN  EACH  OF  THE  ABOVE 
VARIATIONS. 

Summary  of  the  methods  of  adjustment. 

Variation  I  (Deficiency),  in  which  the  adjustment  is  made 
by  an  additional  annual  instalment  to  he  set  aside  during  the 
whole  of  the  U7iexi)ired  portion  of  the  original  repayment  period. 

Statement  XVI.  A. 

(-7)  Ascertain  the  amount  of  the  present  deficiency  and 
calculate  the  equivalent  amount  of  original  loan  by  one 
of  the  methods  described  in  Chapter  XV . 

Calculation  {XV)  6.    £734-659. 

(2)  Calctilate  the  additional  annual  sinking  fund  instalTuent 

to  be  set  aside  and  accuTnulated  for  the  whole  of  the 

unexpired  portion  of  the  original  repayment  period  to 

provide  the  above  equivalent  amount  of  original  Loan. 

Calculations  {XV)  3  and  (XVT)  1.    £45-594. 


172    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

(J)  The  additional  annual  instalment  so  ascertained  added  to 
the  original  annual  instalment  will  give  the  augviented 
annual  instalinent  to  he  set  aside  during  the  whole  of 
the  unex'pired  'portion  of  the  repayment  period. 

(4)  Prepare  a  statement  showing  the  final  repayment  of  the 

loan  by  the  operation  of  the  fund  under  the  amended 
conditions.  Statement  XVI.  A. 

id)  Prepare  a  pro  forma  account  showing  the  amount  which 
should  he  in  the  fund  at  the  end  of  each  year  of  the 
unexpired  portion  of  the  repayment  period  for  after 
reference .  Pro  forma  Account,  No.  2. 

Variation  II  (Deficiency),  in  whicli  the  adjustment  is  made 
hy  an  additional  annual  instalment  (£104'039)  to  he  set  aside 
during  part  only  (-5  years)  of  the  unexpired  portion  [13  years) 
of  the  original  repayTnent  period  {25  years). 

Statement  XYI.  C. 

Note.  In  order  to  Tnake  tJie  following  summary  perfectly 
clear  it  contains  (^in  hracl-ets)  the  results  ascertained  in  the 
example  afterwards  tcorked  out  in  detail.  The  anriual  incre- 
ment (ratio)  method,  previously  referred  to  in  Chapter  XIV, 
cannot  he  applied  to  cases  in  loliich  the  amended  instalment  is 
not  spread  equally  over  the  whole  of  the  period. 

(i)  Ascertain  the  a'mount  of  the  present  deficiency  {£469'744) 
and  calculate  the  equivalent  amoxint  {£734'659)  of 
original  loan,  as  described  in  Chapter  XV . 

Calculation  {XV)  6. 

(2)  Divide  the  unexpired  portion  {13  years)  of  the  original 
repayment  period  {25  years)  into  two  parts,  as  follows  :  — 

1st  portion  {5  years),  during  udiich  the  additional 
annual  instalment  is  required  to  he  set  aside. 

2nd  portion  {8  years),  during  uliich  the  additional 
annual  instalment  is  not  required,  to  he  set  aside,  hut  only 
the  annual  instalment  as  originally  ascertained. 

(5)  Calculate    the    present    value    {£557'908)    of    the    above 

equivalent  amount  {£734'659)  of  the  original  loan,  as  if 
it  were  due  at  the  end  of  a  number  of  years  {8)  equal  to 
the  second  portion  of  the  unexpired  repayment  period 
{13  years).  Calculation  {XVI)  3. 


THE    CORRECTION    OF    A    DEFICIENCY  173 

(4)  Calculate  the  additional  annual  instalment  (£104'039)  to 

be  set  aside  and  accumulated  for  a  numher  of  years  (5)  in 
the  first  'portion  of  the  unexpired  repayment  period'  [13 
years)  to  provide  the  present  value  {£5o7'90S)  so  found,  as 
above.  Calculation  {XVI)  4. 

(5)  The   additional  annual   instalment   so   found   {£104'039) 

added  to  the  original  annual  instalment  {£680'234)  icill 
give  the  -augmented  annual  instalment  [£784'273)  to  be 
set  aside  during  the  first  portion  (5  years)  of  theunea-pircd 
repayincnt  period  (13  years). 

(6)  The  original  annual  instalment  [£680'234)  tvill  continue 

to  be  set  aside  and  accumulated  during  the  second  portion 
[8  years)  of  the  unea-pired  repayment  period'  (13  years). 

(7)  Prepare  a  statement  shoicing  the  final  repayment  of  the 

loan  by  the  operation  of  the  fund  binder  the  amended 
conditions.  Statement  XVI.  C. 

(8)  Prepare  a  pro  forma  account  showing  the  amount  which 

should  be  in  the  fund  at  the  end  of  each  year  of  the 
unexpired  period  for  reference  in  after  years. 

Pro  forma  Account,  Xo.  3. 

Xote.  The  calculations  in  this  and  subsequent  chapters  itill 
be  found  in  the  Appendix,  but  each  calculation  will  be  shown 
by  only  one  of  the  three  methods  given  in  the  stcmdard  forms. 

Variation  I.  The  correction  of  a  deficiency  in  tlie  fund  by  an 
additional  annual  instalment  to  be  set  aside  during-  tlie 
whole  of  the  unexpired  portion  of  the  repayment  period. 

Statement    XAI.    A. 

In  Chapter  XY,  the  factors  relating  to  a  deficiency  in  a 
sinking  fund  have  been  fully  discussed,  and  several  methods 
described  by  which  to  ascertain  the  resulting  additional  annual 
instalment  to  be  spread  equally  over  the  whole  of  the  unexpired 
portion  of  the  repayment  period.  Two  alternative  methods 
have  been  pointed  out  by  which  the  deficiency  may  be  corrected, 
both  of  which  agree  in  providing  an  additional  annual  instal- 
ment, but  differ  as  to  the  number  of  years  over  which  such 
increased  contributions  shall  be  spread.  Sound  finance  demands 
that  the  error  should  be  put  right  by  an  immediate  payment 
of  the  deficiency  into  the  fund,  or  that  the  increased  annual 
contribution  should  be  spread  over  a  shorter  term  than  the  full 
unexpired  portion  of  the  original   repayment  period,   but   th:-- 


174    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

circumstances  of  individual  cases  may  render  it  more  equitable, 
or  perhaps  more  convenient,  tliat  the  adjustment  should  be 
spread  over  the  longest  possible  period. 

The  present  deficiency  of  £469' 744  if  immediately  paid  into 
the  fund  and  accumulated  until  the  end  of  the  period,  will  then 
provide  £734"659  of  original  loan  which  would  otherwise  have 
been  unprovided  for.  The  additional  annual  instalment  of 
£45"594,  to  be  set  aside  and  added  to  the  fund  during  the  whole 
of  the  unexpired  period,  has  already  been  ascertained  by 
Calculation  (XY)  3,  and  it  will  now  be  proved  by  a  further 
Calculation  (XYI)  1  upon  the  authors  standard  calculation 
form  No.  3x,  based  upon  Table  III,  which  is  the  usual  method 
of  finding  the  sinking  fund  instalment.  This  and  other 
calculations  subsequently  referred  to  will  l)e  found  in  the 
appendix. 

Having  ascertained  the  required  additional  annual  instal- 
ment, it  is  now  possible  to  review  the  operation  of  the  fund  so 
amended  in  order  to  show  the  final  repayment  of  the  loan  by  the 
following  Statement  XYI,  A.,  which  will  apply  equally  to  all 
similar  cases  of  adjustment  due  to  a  deficiency  in  the  fund, 
irrespective  of  the  method  by  which  the  additional  annua] 
instalment  is  ascertained  provided  that  such  additional  annual 
instalment  be  spread  equally  over  the  whole  of  the  unexpired 
portion  of  the  repayment  period. 

The  following  Statement  XYI.  A.  also  shows  that  the 
present  investments  of  £9,463  will,  if  accumulated  at  3^  per 
cent,  until  the  end  of  the  period,  then  provide  for  the  repayment 
of  £14799-71  of  original  loan.  Before  making  the  above 
correction  the  balance  of  the  loan  unprovided  for  was  repre- 
sented by  :  — 

The    remaining    original    annual    instalments    of 
£680'234  and  their  accumulations 

Calculation  (XY)  5       £10960-62 
The  deficiency  at  the  end  of  the  12th  year  £469-74 
and    the    loss    of    accumulated    interest 

ca used  therebv 26493 

.  £734-67 


Balance  of  Loan £11695-29 


After  making  tlio  above  adjustment  this  amount  will  be  provided 
by  the  accumulation  of  the  augmented  annual  instalment  of 
£725-828,  as  shown  by  Calcuhition  (XYI)  2. 


THE    CORRECTION    OF    A    DEFICIENCY  175 

The  Annual  Increment  (balance  of  loan)  Method.     The 
annual  increment  has  been  fully  described  in   Chapter  XIV, 
where  it  is  shown  that  it  may  be  used  to  simplify  the  majority 
of  the  adjustments  in  a  sinking  fund,  rendered  necessary  by  any 
variation  from  the  original  conditions  as  to  the  repayment  of 
the  loan.     There  is,  however,  one  limitation,  namely,  that  any 
variation  in  the  rate  of  income  to  be  received  upon  the  present 
investments,  or  in  the  rate  of  accumulation,  must  apply  equally 
to  the  whole  of  the  future  period  of  repayment,  which,  however, 
may  be  increased  or  reduced.     It  is   also  necessary  that  any 
increased  or  reduced  annual  instalment,  consequent  upon  any 
such  variation  in  the  original  conditions,  shall  be  spread  equally 
over  the  whole  of  the  unexpired  portion  of  the  repayment  period. 
For  this   reason,   therefore,    the   method   has   been   applied    in 
Statement  XVI.  B.  to  the  foregoing  example  (Variation  I)  in 
which  the  deficiency  of  £469' 744  is  made  good  by  an  additional 
annual  instalment  of  £45"594,   to  be  spread  equally  over  the 
whole  of  the  unexpired  portion  of  the  repayment  period,  but  the 
method    will    not    apply    to    the    example    following,    namely, 
Variation   II,    in   which   the  additional   annual    instalment   is 
required  to  be  spread  over  the  earlier  years  only  of  such  un- 
expired term.     If  this  method  be  applied  to  the  latter  example 
the  result  would  be  only  the  equated  annual  instalment,  which, 
however  interesting  from  a  theoretical  point  of  view,  would  not 
be   of   any   practical    use    under   the    actual    conditions.        An 
example  of  an  equated  annuity  is  given  and  fully  described  in 
Chapter  XXVII. 

This  method  of  making  the  adjustment  of  a  sinking  fund  by 
means  of  the  annual  increment  is  practically  the  same  as  that 
adopted  in  the  case  of  local  authorities,  where  the  whole  of  the 
annual  instalments,  as  and  when  set  aside,  are  immediately 
applied  in  the  actual  repayment  of  debt.  Section  234  (5)  of  the 
Public  Health  Act,  1875,  provides  that  where  any  part  of  the 
fund  is  so  applied  there  shall  be  paid  into  the  fund  and  charged 
to  the  rate  account  the  interest  which  would  have  been  earned  by 
the  part  of  the  fund  so  applied.  If  it  be  assumed  that  the 
whole  of  the  fund  is  so  applied  in  repayment  of  the  debt,  and  the 
rate  of  interest  payable  upon  the  loan  is  the  same  as  the  rate  of 
accumulation  of  the  fund,  the  amount  charged  annually  to  rate 
account  in  respect  of  interest  and  redemption  charges,  is  the 
annual  increment  of  the  fund,  using  the  term  in  the  sense  here 
applied  to  it. 


176    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


A  Deficiency  in  the  Fund,  Statement  XVI,  A. 

Showing  the  final  repayment  of  the  Loan,  by  the  operation 
of  the  sinking  fund,  after  making  the  adjustment  in  the 
annual  instalment,  consequent  ujion  a  deficiency  in  the 
amount  which  should  stand  to  the  credit  of  the  fund. 

Variation  I  (Deficiency),  in  which  the  additional  annual 
instalment  is  set  aside  during  the  whole  of  the  unexpired 
portion  of  the  repayment  period. 


Present  investments  (at  end  of  12thyear),£946300 


Equivalent 

amount  of 

original  loan. 


Amount  thereof,  accumulated  for  13  years  at 

3i  per  cent.  Calculation  (XV)  4  £14799-71 

Amended  annual  instalment : — 

Original  annual  instalment     £680'234 

Additional  annual  instalment 

Calculation  (XVI)  1        45-594 


£725-828 


Amount  thereof  in  13  years  at  3|  per  cent. 

Calculation  (XVI)  2  £11695-29 


Amount  of  original  loan £26495-00 


THE    CORRECTION    OF    A    DEFICIENCY  177 

A  Deficiency  in  the  Fund.  Statement  XVI,  B. 

The  Annual  Increment  (balance  of  loan)  Method. 

To  find  the  amended  annual  sinking  fund  instalment  consequent 
upon  a  deficiency  in  the  amount  which  should  stand  to 
the  credit  of  the  fund. 

Variation  I  (Deficiency),  in  which  the  additional  annual 
instalment  is  set  aside  during  the  whole  of  the  unexpired 
portion  of  the  repayment  period. 

Amount  of  originalloan  (25  years) £2649500 

deduct  amount  in  the  fund  at  the  end  of  the 

12th  year       £9463-00 


Balance  of  loan    £17032-00 


Amended  annual  increment  to  be  added  to  the  fund 
and  accumulated  at  3^  per  cent.,  to  provide 
this  amount  at  the  end  of  13  years. 

Calculation  (XYI)  9     £1057-033 

deduct  income  to  be  received  from  the  present 

investments,  £9,463,  at  3^  per  cent.       £331-205 


Amended  annual  instalment,  feemg'        £725-828 

Original  annual  instalment  ...  £680-234 

Additional  annual  instalment     £45-594 

£725-828 


The  rule  relating  to  this  method    is  stated  at  the  head  of 
Chapter  XXII. 


178 


REPAYMENT   OF   LOCAL  AND   OTHER   LOANvS 


Pro  forma  Sinking  Fund  Account,  No.  2. 

A  Deficiency  in  the  Fund.     (Variation  I.) 
Loan  of  £26,495,  repayable  at  the  end  of  25  years. 

Showing  the  final  repayment  of  the  loan,  by  tlie  operation  of 
tlie  amended  annual  instalment  of  £725'828,  to  be  set  aside 
during  the  whole  of  the  unexpired  period  of  repayment. 

Statement  XVI.  A.  E-ate  of  accumulation,  3^  per  cent. 


Year. 

Amount  in 

the  fund 

at  beginning 

of  year. 

Income 

received  from 

investments 

3i  per  cent. 

Annual 
sinking  fund 
instalment. 

Amount  in 

the  fund 

at  end 

of  year. 

Year. 

1 

1 

2 

2 

3 

3 

4 

The  amount  in  the 

fund  at  the  end  of  the 

4 

5 

12th  year,  £9,463,  is  an  assumed  amount, 

5 

6 

and    is 

equivalent 

to    setting 

aside    an 

6 

7 

annual  instalment  of  £648'064, 

as  shown 

7 

8 

by  Calculation  (XVI)  10,  instead  of  the 

8 

9 

correct  annual  instalment  of  £680"234. 

9 

10 

10 

11 

11 

12 

9463000 

12 

13 

9463000 

331-205 

725-828 

10520-033 

13 

14 

10520033 

368-201 

725-828 

11614-062 

14 

15 

11614062 

406-492 

725-828 

12746-382 

15 

16 

12746-382 

446-123 

725-828 

13918-333 

16 

17 

13918-333 

487-142 

725-828 

15131-303 

17 

18 

15131-303 

529-596 

725-828 

16386-727 

18 

19 

16386-727 

573-535 

725-828 

17686090 

19 

20 

17686090 

619-013 

725-828 

19030-931 

20 

21 

19030-931 

666-083 

725-828 

20442-842 

21 

22 

20442-842 

714-799 

725-828 

21863-469 

22 

23 

21863-469 

765-221 

725-828 

23354-518 

23 

24 

23354-518 

817-408 

725-828 

24897-754 

24 

25 

24897-754 

871-418 

725-828 

26495-000 

25 

THE    CORRECTION    OF    A    DEFICIENCY  179 

Yariation  II.  The  correction  of  a  deficiency  in  the  fund  by 
an  additional  annual  instalment,  to  be  set  aside  during 
the  earlier  years  only  of  the  unexpired  portion  of  the 
repayment  period.  Statement  XYI.  C. 

The  correction  of  the  deficiency  in  this  manner  is  more 
complicated  than  by  spreading  the  additional  annual  instalment 
equally  over  the  whole  of  the  unexpired  portion  of  the  repayment 
period,  but  is  not  at  all  difficult.  The  factors  immediately 
concerned  are  (1)  the  present  deficiency  of  £469"T44;  (2)  the 
amount  of  original  loan  £734'659,  represented  by  such  deficiency, 
and  (3)  the  original  annual  instalment  of  £680'234. 

In  the  present  example  it  will  be  assumed  that  the  additional 
annual  instalment  is  required  to  be  of  such  increased  amount  (as 
compared  with  the  additional  annual  instalment  of  £45"594  to 
be  spread  over  the  whole  of  the  unexpired  period)  that  it  will 
be  sufficient  to  make  up  the  present  deficiency  if  set  aside  for 
5  years  only,  instead  of  for  13  years.  Under  this  alternative 
method  the  unexpired  period  of  13  years  is  divided  into  two 
parts.  During  the  first  five  years  the  additional  annual 
instalment  will  be  set  aside  and  accumulated  at  3^  per  cent,  in 
augmentation  of  the  original  annual  instalment.  At  the  end 
of  the  five  years  this  additional  annual  instalment  will  cease, 
and  will  then  have  amounted  to  a  sum  which  will  continue  to 
accumulate  at  compound  interest  for  a  further  eight  years. 
The  accumulated  amount  of  the  additional  annual  instalment  at 
the  end  of  five  years,  should,  at  the  end  of  the  remaining  eight 
years,  amount  to  the  balance  (£734'659)  of  loan  not  otherwise 
provided  for.  The  adjustment  may  be  made  by  direct  calcula- 
tion, and  may  also  be  made  by  steps.  A  similar  method  by 
step  has  been  adopted  Avhen  dealing  with  a  variation  in  the 
future  rate  of  income  to  be  received  upon  the  present  invest- 
ments when  it  is  known  in  advance  that  such  a  variation  will 
take  effect  at  a  definite  future  date  during  the  unexpired  portion 
of  the  redemption  period,  as  explained  in  Chapter  XXYII.  In 
order  to  determine  the  additional  annual  instalment  to  be  set 
aside  and  accumulated  for  the  first  period  of  five  years,  it  is 
first  necessary  to  ascertain  the  sum  to  which  it  is  required  to 
accumulate  at  the  end  of  five  years,  which  latter  sum  will  in  its 
turn  accumulate  without  further  addition  for  a  further  period 
of  eight  years.  At  the  end  of  the  unexpired  period  of  13  years 
it  is  necessary  to  provide  £734" 659,  and  the  first  step  is  to 
ascertain  the  sum  which,  if  accumulated  at  3^  per  cent,  for 
eight  years,  will  amount  to  £734' 659 ;  in  other  words,  to  find  the 
present  value  of  £734"659  under  the  above  conditions,  namely, 


iSo    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

£557-908.  Cak'ulatiou  (XYI)3.  The  next  step  is  to  ascertain 
the  annual  instalment  which  will  amount  to  £557'908  if  set 
aside  and  accumulated  at  3|  per  cent,  for  five  years. 

This  is  a  similar  problem  to  the  previous  one  dealing  with 
the  present  deficiency  of  £469744,  where  it  was  required  to  find 
the  annual  instalment  to  amount  to  £734"659,  Calculation  (XYI) 
1,  and  also  similar  to  Calculation  (XY)  1  required  to  find  the 
original  annual  instalment  of  £680'2:')4.  Calculation  (XYI)  4 
shows  that  the  equal  annual  instalment  to  provide  £557 '908  at 
the  end  of  five  years  at  3^  per  cent,  is  £104-039.  The  method 
of  complying  with  the  above  conditions  has  noAv  been  ascertained. 
It  has  been  found  by  Calculation  (XYI)  4  that  an  annual 
instalment  of  £104-039  set  aside  for  five  years  and  accumulated 
at  3^  per  cent,  will  at  the  end  of  that  time  amount  to  £557-908, 
and  it  has  been  found  by  Calculation  (XYI)  3  that  this  sum  of 
£557-908,  accumulated  at  3^  per  cent,  for  eight  years,  will 
amount  to  £734-659,  which  is  the  portion  of  the  original  loan 
not  otherwise  provided  for,  owing  to  the  present  deficiency  of 
£469-744. 

The  sinking  fund,  as  amended  by  the  results  of  the  foregoing 
calculations  will  now  consist  of  :  — 

A  present  credit  to  the  fund,  represented  by  invest- 
ments valued  at ^ £9463-000 


An  augmented  annual  instalment  for  5  years  made 
up  as  follows  : 

Original  instalment     £680-234 

Additional  instalment  for  5  years      104-039 

■        £784-273 

The  original   annual  instalment  to  be  continued 

for  a  further  8  vears  of    £680-234 


And  the  above  provision  accumulated  at  3^  per  cent.,  as 
originally  caktriated,  will  at  the  end  of  the  prescribed  period  of 
repayment,  namely,  25  years,  be  sufficient  to  provide  the  full 
amount  of  the  original  loan  of  £26,495. 

In  order  to  complete  the  argument  it  is  necessary  to  show 
the  position  of  the  fund  at  the  end  of  the  17th  year  when  the 
additional  annual  instalment  of  £104039  will  cease  and  to 
continue  the  accumulation  of  the  fund  from  that  time  until  the 
end  of  the  original  term  of  25  years.  During  the  second  period 
of  eight  years,  as  previously  mentioned,  the  original  instalment 


THE    CORRECTION    OF    A    DEFICIENCY  i8i 

uf  £680'234  only  will  continue  to  be  set  aside  and  added  to  the 
fund.  The  following  Statement  XYI.  C.  shows  the  final 
repayment  of  the  loan  by  the  operation  of  the  fund  after  making 
the  above  adjustment. 

In  the  foregoing  statement  a  break  has  been  made  at  the  end 
of  the  17th  year,  being  the  end  of  the  five  years  during  which 
the  corrective  instalment  of  £104089  is  required  to  be  set  aside. 
The  calculation  might  have  been  simplified  by  ascertaining, 
in  the  direct  manner  shown  in  Statement  XYI.  D.l,  the 
amount  of  loan  Avhich  will  be  provided  by  the  accumulation  at 
the  end  of  13  years  of  the  instalment  of  £104'0-j9  to  be  set  aside 
for  five  years  only.  This  direct  method  by  step  is  f ulh'  explained 
in  Chapter  XXYII,  Statement  C,  where  it  is  applied  to  find 
the  amount  of  loan  which  Mnll  be  provided  by  the  accumulation 
of  the  income  from  the  present  investments,  such  income  being 
at  varying  known  rates  per  cent,  during  the, unexpired  period. 
(The  calculation  might  also  have  been  made  in  terms  of  the 
amended  annual  instalment  of  c£T84'2T3).  In  conclusion,  a 
further  Statement  XYI.  D".2,  has  been  prepared,  showing  the 
final  repayment  of  the  loan,  which  should  be  compared  with 
Statement  XYI.  C,  in  order  to  show  the  simplification  of  the 
proof  by  the  method  by  stej?. 


i82    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


A  Deficiency  in  the  Fund.  Statement  XVI.  C. 

Showing  the  final  repayment  of  the  loan,  by  the  operation  of 
the  sinking  fund  after  making  the  adjustment  in  the 
annual  instalment  consequent  upon  a  deficiency  in  the 
amount  which  should  stand  to  the  credit  of  the  fund. 

Variation  II  (Deficiency),  in  which  the  additional  annual 
instalment  is  set  aside  during  the  earlier  part  only  of  the 
unexpired  portion  of  the  repayment  period. 

Equivalent 

amount  of 

original  loan. 

Present  investments  (at  end  of  12th  year),  £946300 


Amount  thereof,   accumulated  for  5  years  at 

31  per  cent.  Calculation  (XYI)  5  £11239-07 

Amended  annual  instalment : — 

Original £680234 

Additional       104039 


£784-273 


Amount  thereof,  accumulated  for  5  years  at 

3i  per  cent.  Calculation  (XYI)  6     £4205-64 


Amount  in  the  fund,  at  end  of  17th  year      £15444-71 


Amount  thereof,   accumulated  for  8  years  at 

31  per  cent.  Calculation  (XVI)  7  £20337-74 

Original  annual  instalment  (resumed)  :— 

Amount  of  £680234  per  annum,  accumulated 
for  8  years  at  3|  per  cent. 

Calculation  (XVI)  8     £6157-26 


Amount  of  original  loan £26495-00 


THE    CORRECTION    OF    A    DEFICIENCY  183 

A  Deficiency  in  the  Fund.  Statement  XVI.  D(l). 

The  Amount  of  {the  Amourit  of  £1  fer  Annum)  Method  by  Step, 
by  Thoinan's  Tables. 

To  find  the  accumulated  amount  of  an  additional  annual  instal- 
ment, or  other  annuity,  to  be  set  aside  and  added  to  the 
sinking  fund  for  a  limited  period  of  years ;  and  at  the  end  of 
that  period  the  accumulated  amount  thereof  to  continue  to 
accumulate  for  a  further  specified  period.  The  rate  of 
accumulation  in  both  periods  may  be  the  same,  or  be  at 
different  rates  per  cent. 

Eequired  the  amount  of  an  additional  annual  instalment  of 
£104-039,  to  be  set  aside  for  a  period  of  5  years,  and 
accumulated  at  3^  per  cent.  At  the  end  of  5  years  the 
annual  instalment  ceases,  but  the  sum  to  which  it  has  then 
amounted  continues  to  accumulate  for  a  further  period  of 
8  years,  also  at  3^  per  cent. 

First  period,  5  years.       Second  period,  8  years. 

Log.  instalment  A^/     104-039     20171984 

acZfZ:  Log.  RN3i  per  cent.     5  years    EN  00747017 

Log.  RN  3i  per  cent.     8  years    RN  0-1195228 


2-2114229 


add  10  to  the  log.  12-2114229 

^efZwrt :  Log-fl*^,  3^percent.     5  years    a^  93453372 

M  2-8660857 

which  is  the  log.   of  the  required  future  amount, 

namely    ^'34-659 

Note.  This  method  may  be  inverted  to  find  the  additional 
annual  instalment  in  the  first  instance  instead  of  as  described 
in  the  text.     See  Statement  XXXIV.  G. 


iS4    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


A  Deficiency  in  the  Fund.  Statement  XVI.  D  (2). 

Showing  the  final  repayment  of  the  loan,  by  tlie  operation  of 
tlie  sinking  fund  after  making  the  adjustment  in  tlie  annual 
instalment,  consequent  upon  a  deficiency  in  the  amount 
which  should  stand  to  the  credit  of  the  fund. 

Variation  II  (Deficiency),  in  which  the  additional  annual 
instalment  is  set  aside  during  the  earlier  part  only  of  the 
unexpired  portion  of  the  repayment  period. 

An  alternative  method  to  Statement  XVI.  C,  based  upon  the 
method  by  step. 

Equivalent 

amount  of 

original  loan. 

Present  investments  (at  end  of  12th  year),£946300 


Amount  thereof,  accumulated  for  13  years  at 

3i  per  cent.  Calculation  (XT)  4  £14799-71 

Original  annual  instalment    £680234 


Amount  thereof,  accumulated  for  13  years  at 

^  per  cent.  Calculation  (XY)  5  £10960-62 

Additional  annual  instalment        £104039 


to  be  set  aside  for  5  years  only,  and  accumu- 
lated  for   a  further   8  years   at   3|   per   cent 
"  Method  by  step  "  Calculation  (XVI)  D.l       £734'67 


Amount  of  original  loan    £2649500 


THE    CORRECTION    OF    A    DEFICIENCY 


185 


Pro  forma  Sinking  Fund  Account,  No.  3, 

A  Deficiency  in  tlie  Fund.     (Variation  II.) 

Loan  of  £26,496,  reijayahJc  at  the  end  of  25  years. 

Showing  the  final  eepayment  of  the  loan,  by  tlie  operation  of 
the  amended  annual  instalment  of  £784-273,  to  be  set  aside 
during  tbe  first  5  years  only  of  the  unexpired  period  of 
repayment. 


Statement  XYI.  C. 


Rate  of  accumulation,  3|  per  cent. 


Year. 
1 

Amount  in 
the  fund 

at  beginning 
of  year. 

Income 

received  from 

investments 

3J  per  cent. 

Amount  in 
Annual                       tlie  fund 
sinking  fund                    at  end 
instalment.                    of  year. 

Year. 
1 

2 

2 

3 

3 

4 

The  amount  in  the  i 

:und  at  the  end  of  the 

4 

5 

12th  year,  £9,463,  is  an  assumed  amount, 

5 

6 

and    is 

equivalent 

to    setting    aside    an 

6 

7 

annual  instalment  c 

)f  £648-064,  as  shown 

7 

8 

by  Calculation  (XVI)  10,  instead  of  the 

8 

9 

correct 

annual  inst; 

alnient  of  £680-234. 

9 

10 

10 

11 

11 

12 

9463-000 

12 

13 

9463-000 

331-205 

784-273         10578-478 

13 

14 

10578-478 

370-247 

784-273         11732-998 

14 

15 

11732-998 

410-655 

784-273         12927-926 

15 

16 

12927-926 

452-477 

784-273         14164-676 

16 

17 

14164-676 

495-761 

784-273         15444-710 

17 

18 

15444-710 

540-567 

680-234        16665-511 

18 

19 

16665-511 

583-293 

680-234        17929-038 

19 

20 

17929-038 

627-516 

680-234         19236-788 

20 

21 

19236-788 

673-288 

680-234        20590-310 

21 

22 

20590-310 

720-661 

680-234        21991-205 

22 

23 

21991-205 

769-692 

680-234        23441-131 

23 

24 

23441131 

820-440 

680-234        24941-805 

24 

25 

24941-805 

872-961 

6S0-234         26495-000 

25 

i86         REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 


CHAPTER  XVII. 

SINKING   FUND    PROBLEMS    RELATING    TO 
THE  AMOUNT  IN  THE  FUND. 

A   surplus    in   the    fund ;    how  it   may  arise,   and  how  it 
may  be  adjusted. 

Variation  I. 

Arising  in  consequence  of  an  excessive  past  accumula- 
tion OF  the  fund. 

Variation  IL 

Arising  in  consequence  of  the  payment  into  the  fund  of 
the  proceeds  of  sale  of  part  of  the  assets  representing 
the  security  for  the  loan,  or  a  realised  profit  upon 

the  SALE  OF  AN  INVESTMENT  REPRESENTING  THE  FUND. 

Statement  XYII.  A. 

Summary  of  the  methods  of  adjustment.  The  various  causes 
leading  to  a  surplus  in  the  fund.  Difference  in 
conditions  and  practice  as  between  local  authorities 
and  commercial  and  financial  undertakings.    comparison 

OF  THE  VARIOUS  METHODS  OF  DEALING  WITH  A  SURPLUS.  ThE 
AN-NUAL  INCREMENT  (BALANCE  OF  LOAN)  METHOD.  STATEMENT 
SHOWING  THE  FINAL  REPAYMENT  OF  THE  LOAN  BY  THE  OPERA- 
TION OF  THE  AMENDED  ANNUAL  INSTALMENT. 

Summary  of  the  methods  of  adjustment. 

Variation  I  (Surplus),  arising  in  consequence  of  an  exces- 
sive 'past  accumulation  of  the  fund. 

(7)  Ascertain  the  actual  present  surplus,  as  described  in 
Chapter  XV . 

{2)  Calculate  the  annuity  or  annual  instalment  of  lohich  this 
sum  is  the  present  value  for  the  unexpired  portion  of  the 
repayment  period.  Similar  to  Calculation  (XV)  3. 


A    SURPLUS    IN    THE    FUND  187 

(t3)  2'he  anmial  instalment  so  ascertained,  deducted  jrora  the 
original  anmial  instalment  loill  give  the  reduced  annual 
instahne7it  to  he  set  aside  during  the  tvhole  of  the  un- 
expired portion  of  the  repayment  period. 

(4)  Prepare  a  state  merit  shoudng  the  firuil  repayment  of  the 

loan  by  the  operation  of  the  sinking  fund  under  the 
amended  conditions.  Similar  to  Statement  XVI.  A., 
with  the  necessary  modifications  relating  to  a  surplus 
instead  of  to  a  deficiency . 

(5)  Prepare  a  pro  forma  amount  shoiving  the  amount  which 

should  be  in  the  fund  at  the  end  of  each  year  of  the 
unexpired  repayment  period. 

Note.  The  above  method  so  closely  resembles  the  one 
adopted  in  the  case  of  a  deficiency  and  the  folloioing  method 
relating  to  Variation  II,  that  no  further  amplification  is 
required.  Unlike  a  d&ficiency,  however,  a  surplus  should  be 
spread  equally  over  the  whole  of  the  unexpired  repa/yment 
period,  and  consequently  Method  II  [Deficiency)  ivill  rarely 
apply. 

Variation  II  (Surplus),  arising  in  consequence  of  the 
payment  into  the  fmid  of  the  proceeds  of  sale  of  part  of  the  assets 
representing  the  security  for  the  loan,  or  a  realised  profit  upon 
the  sale  of  an  investment  representing  the  fund. 

Statement  XVII.  A. 

(7)  Ascertain  in  the  manner  described  in  Chapter  XV, 
whether  there  is  a  surplus  or  a  deficiency  in  the  fund 
apart  from  the  proceeds  of  realisation  now  under  con- 
sideration; and  if  so,  calculate  the  corrective  annual 
sinking  fund  instalment  required.        Calculation  {XV)  3. 

(2)  Calculate  the  annuity  which  may   be  purchased  for  the 

unexpired   portion   of    the   repayment   period,    with    the 
amount  now  paid  into  the  fund.        Calculation  [XVII)  1. 
[Here  refer  to  memo,  after  (^).] 

(3)  Deduct   the   annuity    so   ascertained  from    the    original 

annual  instalment,  and  adjust  the  latter  also,  if  required, 
by  the  above  corrective  instalment,  referred  to  in  [1). 

(4)  The  remainder  will  be  the  future  reduced  annual  sinking 

fund  instalment,  to  be  set  aside  and  accumulated  durmg 
the  whole  of  the  unexpired  portion  of  the  repoAfment 
period. 


i88    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

(<5)  Prepare  a  statement  shoiciny  the  final  repayment  of  the 
loan  hrj  the  operation  of  the  sinliing  fund  under  the 
amended  conditions.  Statement  XVII.  A. 

(6)  Prepare  a  pro  forma  account  showing  the  amount  which 
sJiould  he  in  the  fund  at  the  end  of  each  year  of  the 
uned'pired  repayment  period.      Pro  forma  Account,  Xo.  4. 

Memo.  If  the  original  annu(d  instalment  he  a  prescribed 
suvi  instead  of  being  found  by  calculation  in  the  ordinary  way 
{see  [XVI)  1),  proceed  by  the  method  described  under  T  ariation 
IV  [Surplus)  in  Chapter  XVIII,  substituting  for  operation  (7) 
in  that  method  the  above  operation  [2). 


A  SuRPLrs  IX  A  Sinking  Fund  and  how  it  may  arise. 

Although  it  does  not  fall  within  the  province  of  a  work  of 
this  character  to  mention  all  the  varions  causes  which  may 
lead  to  the  existence  of  a  surplus  in  a  sinking-  fund,  vet  it  is 
very  advisable  to  give  a  brief  outline  of  the  princi})al  ways  in 
which  this  may  happen,  and  which  may  be  divided  into  the 
following  classes,  any  two  or  more  of  which  may  operate 
simultaneously  :  — 

(1)  An  excess  in  the  amount  of  the  annual  instalments  pre- 

viously paid  into  the  fund  or  an  increase  in  the  rate  of 
accumulation  in  excess  of  the  rate  assumed  in  calculating 
the  original  instalment.  Variation  I. 

(2)  The  payment  into  the  fund  of  a  realised  profit  upon  the 

sale  of  an  investment  representing  the  fund  or  the 
proceeds  of  sale  of  part  of  tlie  assets  representing  the 
security  for  the  loan.  Variation  II. 

(3)  In  the  case  of  commercial  or  financial  undertakings,  there 

may  be  a  change  in  the  character  of  part  of  the  original 
loan,  whereby  the  original  obligation  to  set  aside  a 
sinking  fund  is  modified  owing  to  the  withdrawal  of  part 
of  the  loan  from  the  operation  of  the  fund. 

Tliis  will  be  fully  discussed  in  the  following  chapter,  where 
it  will  be  shown  that  the  precise  metliod  of  making  the  adjust- 
ment depends   upon  the  nature  of  tlu'  original  annual  instal- 


A    SURPLUS    IN    THE    FUND  189 

ment,    nnd    the    problem    will    be    divided    into    two    parts    as 
f olloAvs  :  — 

A.  In  wliicli  the  original  annual  instalment  was  found  by 

calculation  based  upon  a  specified  period  of  repayment 
and  rate  of  accumulation.  Variation  III. 

B.  In  which  the  original  annual  instalment  is  a  stated  sum 

and  is  not  based,   except   in   a  general  way,   upon   any 
period  of  repayment  or  rate  of  accumulation. 

Variation   IV. 

Variation  I  (Surplus),  arising  in  consequence  of  an  excessive 
past  accumulation  of  the  fund. 

This  will  be  of  rare  occurrence  if  the  pro  forma  account 
already  recommended  has  been  made  out  showing  the  operation 
of  the  fund  until  maturity,  and  any  such  minor  instances  may 
be  adjusted  as  and  when  they  arise  by  transfers  to  the  current 
year's  rate  or  revenue  account.  In  the  case  of  larger  amounts 
they  may  be  treated  in  the  manner  mentioned  in  Chapters  XV 
and  XVI,  referring  to  a  deficiency  in  the  fund,  but  of  course  by 
reducing  the  future  annual  instalment. 

Variation  II  (Surplus),  arising  in  consequence  of  the  payment 
into  the  fund  of  the  proceeds  of  sale  of  part  of  the  assets 
representing  the  security  for  the  loan  or  a  realised  profit 
upon  the  sale  of  an  investment  representing  the  fund. 

Statement  XVII.  A. 

This  chapter  will  deal  fully  with  those  cases  in  which  the 
sinking  fund  obligations  are  modified  by  the  payment  into  the 
fund  of  the  proceeds  of  sale  of  part  of  the  security  for  the  loan 
to  be  ultimately  repaid,  and  attention  will  be  directed  to  the 
difference  in  practice  as  between  local  authorities  and  com- 
mercial and  financial  undertakings.  In  the  case  of  a  local 
authority  the  sinking  fund  instalment  is  set  aside  to  repay  the 
loan  at  the  end  of  the  period  allowed  under  the  general  or 
special  Act.  These  loans  are  invariably  expended  upon  works 
of  a  capital  nature,  and  it  sometimes  happens  that  part  of  the 
property  representing  the  security  for  the  loan  is  sold.  The 
practice  generally  followed  in  the  case  of  local  authorities  is  to 
pay  such  proceeds  into  the  fund  and  apply  the  same  in  the 
redemption  or  repayment  of  part  of  the  original  loan.  This  is 
as  it  shoiild  be,  and  is  the  practice  adopted  in  commercial  and 
financial  undertakings,  but  it  has  an  important  effect  upon  the 
sinking  fund  instalment.     The  repayment,   during  the  period 


igo         REPAYMENT   OF   LOCAL  AND   OTHER   LOANvS 

allowed,  of  part  of  tlie  debt  out  of  tlie  proceeds  of  sale  of  part 
of  the  security  (instead  of  out  of  the  sinking  fund  provided  out 
of  current  rates  or  profits)  anticipates  the  natural  effect  of  the 
sinking  fund,  and  by  reducing  the  loan  repayable  at  the  end  of 
the  period  correspondingly  reduces  the  necessity  to  set  aside  in 
future  the  full  original  sinking  fund  instalment. 

It  is  obvious  therefore  that  the  original  annual  sinking  fund 
instalment  niay  be  reduced  during  the  remainder  of  the  term  to 
such  an  amount  as  will  provide  the  balance  of  the  debt  not 
repaid  out  of  the  proceeds  of  the  sale  of  part  of  the  assets. 

This  principle  is  followed  in  the  case  of  commercial  and 
financial  undertakings,  but  in  the  case  of  local  authorities  the 
Local  Government  Board  may  require  that  the  proceeds  of 
such  sales  shall  be  paid  into  the  sinking  fund,  and  that  the 
full  amount  of  the  original  annual  instalment  shall  continue  to 
be  set  aside.  The  effect  of  this  is  to  shorten  the  original 
period  allowed  for  the  redemption  of  the  debt.  There  is  not 
any  objection  to  this  method  except  that  the  result  is  to  relieve 
the  later  generation  of  ratepayers  at  the  expense  of  the  present, 
but  in  its  favour  is  the  fact  that  it  is  always  sound  finance  to 
repay  debt  as  soon  as  possible.  In  the  case  of  commercial  and 
financial  undertakings  the  practice  varies,  depending  in  each 
instance  upon  the  conditions  laid  down  in  the  deed  relating 
to  the  loan.  Generally  speaking,  it  may  be  considered  equitable 
in  the  case  of  such  undertakings  to  reduce  the  sinking  fund 
instalment  and  so  maintain  the  original  period  allowed  for  the 
repayment  of  the  debt.  In  the  case  of  a  debenture  stock  repay- 
able on  a  fixed  future  date  this  would  necessarily  require  to  be 
so  unless  j^art  of  the  stock  were  redeemed  by  purchase  upon  the 
open  market. 

The  proceeds  of  sales  of  capital  assets  forming  part  of  the 
security  would,  failing  actual  redemption,  be  invested  in 
securities  authorised  by  the  deed,  and  the  resulting  income 
would  be  added  to  the  sinking  fund  during  the  unexpired 
portion  of  the  repayment  period,  and  therefore  the  future 
annual  instalments  to  be  provided  out  of  the  profits  of  the 
undertaking  would  be  correspondingly  reduced.  If,  however, 
in  the  case  of  a  commercial  or  financial  undertaking  any  such 
proceeds  arising  from  the  sale  of  part  of  the  security  were 
actually  applied  in  redemption  of  part  of  the  loan,  instead  of 
being  invested  in  outside  securities,  the  profit  and  loss  account 
of  the  undertaking  would  be  relieved  to  the  extent  of  the  annual 
interest  payable  upon  such  redeemed  debt,  but  the  sinking  fund 
-would  not  then  be  increased  by  any  income  arising  from  the 


A    SURPLUS    IN    THE    FUND  191 

investment.  Another  difference  between  the  sinking  funds  of 
commercial  and  financial  undertakings  and  those  of  local 
authorities  arises  from  the  fact  that  in  the  former  the  annual 
instalment  is  not  always  charged  against  the  profits  of  the 
undertaking  but  may  be  taken  out  of  the  general  assets  of  the 
concern. 

The  method  of  adjustment  will  be  illustrated  by  the  follow- 
ing example  relating  to  a  commercial  or  financial  undertaking. 

A  sinking  fund  has  been  set  aside  and  accumulated  to 
provide  for  the  repayment  of  a  loan  of  £26,495  at  the  end  of 
25  years — and  in  fixing  the  annual  instalment  the  rate  of 
accumulation  was  taken  at  3^  per  cent.  At  the  end  of  the 
12th  year  the  fund  stands  at  the  proper  estimated  amount 
shown  by  the  pro  forma  account,  and  as  found  by  Calculation 
(XV)  2,  namely,  £9932'T44.  At  that  time  a  portion  of  the 
assets  (forming  part  of  the  security  for  the  loan)  is  realised,  and 
produces,  say,   £4,560. 

The  trust  deed  provides  that  this  amount  shall  be  paid  into 
the  sinking  fund  and  invested,  and  accumulated  until  the  loan 
is  repayable,  namely,  at  the  end  of  the  25th  year,  and  that  the 
future  annual  sinking  fund  instalments  may  be  correspondingly 
reduced.  In  the  present  example  there  is  not  any  question  of 
the  rate  of  income  on  the  present  investments,  or  the  future  rate 
of  accumulation,  being  less  than  3^  per  cent., the  rate  originally 
assumed  in  calculating  the  annual  instalment.  The  effect  of 
the  realisation  of  part  of  the  security  for  the  loan  is  that  the 
amount  in  the  sinking  fund  is  suddenly  increased  by  the  sum 
of  £4,560,  which  was  not  anticipated  when  the  original  annual 
instalment  was  calculated.  If  therefore  this  amount  be  paid 
into  the  sinking  fund  and  accumulated,  and  the  original 
instalments  continue  to  be  set  aside  in  future  and  paid  into  the 
fund  until  the  end  of  the  25th  year,  the  sinking  fund  will  at  the 
end  of  that  period  be  in  excess  of  the  amount  required  to  repay 
the  loan,  and  the  excess  will  be  the  amount  of  the  above  sum 
of  £4,560  accumulated  at  3|  per  cent.,  compound  interest,  for 
13  years. 

The  method  of  ascertaining  the  amount  by  which  the 
original  annual  instalment  may  be  reduced  during  the  un- 
expired portion  of  the  repayment  period  is  exactly  similar  in 
principle  to  that  adopted  in  the  case  of  the  deficiency  of 
£469'744  in  Chapter  XY.  In  that  case  the  deficiency  was 
converted  into  terms  of  original  loan  and  the  annual  instalment 
to  be  set  aside  during  the  remaining  13  years  to  redeem  the 
portion  of  the  loan  not  already  provided  for  was  ascertained. 


192  REPAYMENT   OF    LOCAL   AND    OTHER   LOANvS 

lu  the  present  case  the  sinking;  fund  stands  at  tlie  proper 
calculated  amount  at  the  end  of  the  12th  year ;  and  the  original 
annual  instalments,  alone,  if  continued  for  a  further  I'S  j-ears, 
Avill  be  amply  sufficient  to  provide  for  the  ultimate  repayment 
of  the  debt.  In  addition  there  is  a  sum  in  hand  of  £4,560, 
which  may  now  be  applied  in  repaying  part  of  the  loan,  and  the 
equivalent  annuity  for  the  remainder  of  the  period  may  be 
applied  in  reduction  of  the  future  annual  instalments  to  be 
added  to  the  fund.  The  £4,560  may  be  regarded  as  a  sum 
which  may  now  be  invested  in  the  purchase  of  an  annuity  for 
l^j  years  on  a  3|  per  cent,  basis.  This  method  is  the  more 
preferable  seeing  that  the  £4,560  is  actually  in  hand,  whereas 
the  deficiency  of  £469"T44  represented  the  present  value  of  a 
sum  due  at  a  future  period  and  was  a  definite  amoimt  only  so 
far  as  it  represented  a  sum  which  should  have  been  in  actual 
possession,  but  was  not  so  in  fact. 

When  discussing  the  adjustment  of  a  sinking  fund  in  the 
case  of  a  deficiency  in  Chapters  XV  and  XYI  several  alternative 
methods  were  pointed  out  depending  upon  the  period  allowed  in 
which  to  make  good  past  deficiencies.  In  the  case  of  the 
surplus  under  review,  there  is  not  any  alternative  to  that  already 
considered  if  the  original  date  of  repayment  be  adhered  to, 
because  the  sum  in  question  is  a  definite  one  and  is  actually  in 
hand.  The  calculation  will  be  made  upon  the  author's  standard 
form  Xo.  5,  relating  to  the  annuity  which  £1  will  purchase. 
It  will  be  seen  that  the  sum  now  paid  into  the  fund  will  effect 
a  decrease  in  the  original  annual  instalment  of  £442" 6008  per 
annum.     Calculation  (XYII)  1. 

The  final  repayment  of  the  loan  by  the  operation  of  the 
amended  instalment  during  the  remaining  13  years  of  the 
original  repayment  period  is  shown  in  the  following  Statement 
XVII.  A. 

The  above  method  should  be  carefully  compared  with  the 
correction  of  a  surplus  in  a  sinking  fund,  caused  by  the  with- 
drawal of  part  of  the  loan  from  the  operation  of  the  fund  owing 
to  the  conversion  of  such  part  of  tlie  loan  into  ordinary  share 
capital.  The  difference  in  the  methods  will  be  fully  described 
in  Chapter  XVIII. 

The  Annual  Increment  (balance  of  loan)  Method.  This 
method  will  now  be  used  for  the  purpose  of  ascertaining  the 
amended  annual  instalment,  based  upon  the  future  annual 
increment,  a  summary  of  which  is  given  at  the  beginning  of 


A    SURPLUS    IN    THE    FUND  193. 

Chapter  XY,  aud  is  fully  described  in  Chapters  XVI  and  XXII. 
As  this  method  is  based  upon  the  same  actual  conditions  as  the 
previous  example,  Statement  XYII.  A.,  showing  the  final 
repayment  of  the  loan,  will  also  apply.  This  method  is  shown 
in  Statement  XVII.  B. 


Comparison  of  the  Methods  of  Dealing  with  a  Surplus 
AND  A  Deficiency  in  a  Sinking  Fund. 

It  is  instructive  to  compare  the  above  results  with  the 
example  worked  out  in  the  case  of  a  deficiency  in  the  fund 
(Variation  I),  seeing  that  both  funds  relate  to  loans  identical  a& 
to  amount,  period  of  repayment,  and  rates  per  cent,  of  income 
and  accumulation. 

In  each  case  also  the  adjustment  is  made  at  the  end  of  the 
12th  year,  and  is  spread  over  the  full  remaining  term  of  13 
years. 

In  the  case  of  the  deficiency  in  the  fund  there  was  an 
ascertained  amount  of  £469" 744  by  which  the  present  invest- 
ments, £9463-00,  fell  short  of  the  amount  of  £9932-744  which 
should  have  been  in  the  fund  in  order  to  carry  out  the  original 
obligation.  This  deficiency  was  corrected  by  setting  aside  an 
additional  annual  instalment,  in  augmentation  of  the  original 
annual  instalment  of  £680-234  during  the  unexpired  portion  of 
the  repayment  period. 

This  instalment  of  £45-594,  which  was  found  by  Calculation 
(XV)  3,  represents  the  annuity  which  might  have  been 
purchased  with  the  above  amount  of  £469-744. 

The  present  surplus  consists  of  an  actual  amount  of  cash^ 
namely,  £4,560,  paid  into  the  fund,  Avhich  is  applied  in 
providing  an  instalment  in  reduction  of  the  original  annual 
instalment  of  £680-234.  This  instalment,  as  found  by  Calcula- 
tion (XVII)  1,  based  on  Table  V,  is  £442-601,  and  represents 
the  annuity  which  might  be  purchased  with  the  above  amount 
of  £4,560  paid  into  the  fund. 

In  both  cases  the  amount,  which  should,  as  shown  by 
Calculation  (XA^)  2,  have  been  to  the  credit  of  the  fund,  is 
£9932-744,  which  amount,  if  accumulated  for  13  years  at  3^  per 
cent.,  would  at  the  end  of  the  period,  as  shown  by  Calculation 
(XVII)  2,  have  amounted  to  £15534-375  of  original  loan. 

In  the  case  of  the  surplus  caused  by  a  payment  into  the  fund 
now  under  consideration,  part  of  the  amount  which  should  be 
in  the  fund  at  the  end  of  the  prescribed  repayment  period  of 
25  vears  is  actually  in  hand  at  the  end  of  the  12th  year,  and 


194         REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

tlierefore  the  future  annual  instalment  must  be  correspondingly 
reduced  owing  to  the  future  accumulation  of  the  sum  of  £4,560 
paid  into  the  fund. 

In  the  two  cases  the  amount  which  should  have  been  to  the 
credit  of  the  fund  at  the  end  of  the  12th  year  was  represented 
as  follows :  — 

In  the  case  of  a  Deficiency  in  the  Fund. 

Amount  at  enfl  Equivalent 

of  year.  amount  of  loan. 

Actual  amount  in  the  fund £946.3-  £14799-71 

Deficiency,    involving    an     additional 

annual  instalment  of  £46-594      ...  469-744  734-67 


£9932-744    £15534-38 


In  the  case  of  a  Surplus  in  the  Fund. 

Amount  at  end  Equivalent 

of  year.  amount  of  loan. 

Actual  amount  in  the  fund £9932-744  £15534-38 

Deficiency.        Nil.                   —  — 


£9932-744    £15534-38 


In  both  cases  the  amount  of  original  loan  to  be  provided  by 
the  accumulation  of  the  future  annual  instalments  for  13  years 
is  the  same,  namely,  £1096062,  being  the  total  of  the  original 
loan,  £26,495,  after  deducting  the  above  amount  of  £15534*38 
already  provided  for. 

The  manner  in  which  this  remaining  portion  of  original  loan 
is  dealt  with  in  the  two  cases  is  shown  in  the  following  table  :  — • 

In  the  case  of  a  Deficiency  in  the  Fund. 

Equivalent 
amount  of  loan. 

Future  original  annual  instalment  of  £680-234,  to 
be  set  aside  and  accumulated  for  13  years  at 
^  per  cent.  Calculation  (XY)  5     £1096062 


A    SURPLUS    IN    THE    FUND 


195 


In  the  case  of  a  Surplus  in  the  Fund. 

Cash  in  hand,  being  proceeds  of  security 
sold  and  added  to  the  fund. 

Calculation  (XVII)  3  £456000       £7131-64 


Future  reduced  annual  instalment  of  £237'633, 
to  be  set  aside  and  accumulated  for  13  years 
at  3i  per  cent.  Calculation  (XYII)  5       £3828-98 


£10960-62 


The  above  future  reduced  annual  instalment  of  £237-633  is 
arrived  at  by  deducting  from  the  original  annual  instalment  of 
£680-234  the  annual  instalment  of  £442601,  which  is  the 
future  equivalent  of  the  capital  sum  of  £4,560  paid  into  the 
fund. 

The  above  tabulated  summary  shows  the  intimate  relation 
between  "  present  value "  and  "  future  amount "  at  the 
beginning  and  end  of  the  same  period  and  at  the  same  rate  per 
•cent.,  and  further  demonstrates  the  connection  between  the 
formulae. 

There  are  here  three  expressions  of  the  value  of  one  and  the 
:same  thing  at  the  same  date,  namely  :  — 


1,  A  sum  in  hand  of 


£4560-00 


2.  A  future  annuity  for  13  years  at  3i  per  cent,  of       £442-601 


3.  A  sum  due  at  the  end  of  that  time  also  at 

3i  per  cent £7131-64 


196    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


A  Surplus  in  the  Fund.  Statement  XVII.  A. 

Showing  the  final  repayment  of  the  loan,  by  the  operation  o£ 
the  sinking  fund  after  making  the  adjustment  in  the 
annual  instalment  consequent  upon  a  surplus  over  the 
amount  which  should  stand  to  the  credit  of  the  fund. 

Variation  II  (Surplus),  arising  in  consequence  of  the  payment 
into  the  fund  of  the  proceeds  of  sale  of  part  of  the  assets 
representing  the  security  for  the  loan,  etc. 

Equivalent 
amount  of 
original  loan. 

Present  investments  (at  end  of  12th  year),  £9932" 74 


Amount  thereof,  accumulated  for  13  years  at 

3i  per  cent.  Calculation  (XYII)  2  £15534-38 

Amount  paid  into  the  fund     £456000 


Amount  thereof,  accumulated  for  13  years  at 

3i  per  cent.  Calculation  (XVII)  3     £7131-64 

Amended  annual  instalment : — 

Original  instalment £680-234 

reduced  by,  (XYII)  4,      442601 


£237-633 


Amount  thereof,  accumulated  for  13  years  at 

31  per  cent.  Calculation  (XYII)  5     £3828-98 


Amount  of  original  loan    £26495-00 


A    vSURPLUS    IN    THE    FUND  197 


A  Surplus  in  the  Fund.  Statement  XVII.  B. 

The  Annual  Increment  (balance  of  loan)  Method. 

To  find  the  amended  annual  sinking  fund  instalment  conse- 
quent upon  a  surplus  over  the  amount  which  should  stand 
to  the  credit  of  the  fund. 

Variation  II  (Surplus),  arising  in  consequence  of  the  payment 
into  the  fund  of  the  proceeds  of  sale  of  part  of  the  assets 
representing  the  security  for  the  loan. 

Amount  of  original  loan  (25  years)        ..£2649500 

deduct  amount  in  the   fund   at  the 

end  of  the  12th  year       ...  £9932-74 

proceeds   of  sale   paid   into 

the  fund    £4560-00 

£14492-74 


Balance  of  loan       £12002-26 


Amended  annual  increment  to  be  added  to  the  fund 
and  accumulated  at  3^  per  cent,  to  provide 
this  amount  at  the  end  of  13  years. 

Calculation  (XYII)  6       £744-879 

deduct  income  to  be  received  from  the  present 

investments,  £14492" 74  at 3^  percent.       £507-246 


Amended  annual  instalment,  being :— £237-633 

Original   annual   instalment    £680-234 

reduced  by  £442601 

237-633 


The  rule  relating  to  this  method  is  stated  at  the  head  of 
Chapter  XXII. 


198 


REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 


Pro  forma  Sinking  Fund  Account,  No.  4, 

A  Surplus    in   tlie    Fund.     (Variation    II.) 

Loan  of  £26,495,  repayable  at  the  end  of  25  years. 

Showing  the  final  repayment  of  the  loan,  by  tlie  operation  of 
the  reduced  annual  instalment  of  £237"633. 


Statement  XYII.  A. 


Rate  of  accumulation,  3|  per  cent. 


Year. 

Amount  in 

the  fund 

at  beginning 

of  year. 

Income 

received  from 

investments 

3i  per  cent. 

Proceeds  of 
Annual      sale  of  assets 
sinking  fund      paid  into 
instalment.        the  fund. 

Amount  in 

the  fund 

at  end 

of  year. 

Year. 

1 

1 

2 

2 

3 

3 

4 

The  i 

imount  in 

the  fund  at 

the  ( 

3nd  of 

4 

5 

the  12th  year,  £9932744,  is 

the  ( 

correct 

5 

6 

calculated  amount,  as  shown 

byO 

alcula- 

6 

7 

tion 

(XY)  2,    and    by    the 

pro 

forma 

7 

8 

9 

10 

account,  No.  1, 

Chapter  XY 

8 

9 

10 

11 

11 

12 

456000C 

)     9932-744 

12 

13 

14492-744 

507-246 

237-633 

— 

15237-623 

13 

14 

15237-623 

533-317 

237-633 

— 

16008-573 

14 

15 

16008-573 

560-300 

237-633 

— 

16806-506 

15 

16 

16806-506 

588-228 

237-633 

— 

17632-367 

16 

IT 

17632-367 

617133 

237-633 

— 

18487-133 

17 

18 

18487-133 

647-050 

237-633 

— 

19371-816 

18 

19 

19371-816 

678-014 

237-633 

— 

20287-463 

19 

20 

20287-463 

710-061 

237-633 

— 

21235-157 

20 

21 

21235-157 

743-230 

237-633 

— 

22216020 

21 

22 

22216-020 

777-561 

237-633 

— 

23231-214 

22 

23 

23231-214 

813-092 

237-633 

— 

24281-939 

23 

24 

24281-939 

849-868 

237-633 

— 

25369-440 

24 

25 

25369-440 

887-927 

237-633 

— 

26495-000 

25 

A    SURPLUS    IN    THE    FUND  199 


CHAPTER  XYIII. 

SINKING  FUND  PROBLEMS,    RELATING   TO 
THE  AMOUNT  IN  THE  FUND. 

A  surplus    in    the    fund,    of    a    commercial    or    fixaxcial 

UNDERTAKING  ARISING  ON  THE  WITHDRAWAL  OF  PART  OF  THE 
LOAN  FROM  THE  OPERATION  OF  THE  FUND,  OWING  TO  THE 
CONVERSION  OF  SUCH  PART  OF  THE  LOAN  INTO  ORDINARY 
SHARE    CAPITAL    OR    STOCK   OF   THE    UNDERTAKING. 

Variation  III,  ix  which  the  original  annual  instal- 
ment WAS  FOUND  BY  CALCULATION  BASED  UPON  A  SPECIFIED 
PERIOD   OF    REPAYMENT    AND    RATE    OF    ACCUMULATION. 

Statement  XVIII.  A. 

Variation  IV,  in  which  the  original  annual  instal- 
ment is  a  stated  sum  and  is  not  based,  except  in  a 
general  way,  upon  any  period  of  repayment  or  rate  of 
accumulation.  Statement  XYIII.  D. 

Summary  of  the  methods  of  adjustment.  Remarks  as  to  the 
sinking  funds  of  commercial  and  financial  undertak- 
INGS.      The  ANTiTUAL  INCREMENT  (BALANCE  OF  LOAN)   METHOD. 

Statement  showing  the  final  repayment  of  the  loan  by 
the  operation  of  the  amended  annual  instalment. 


Summary  of  the  methods  of  adjustment. 

Variation  III  (Surplus),  arising  on  the  ivithdrawal  of  lyart 
of  the  loan  from  the  operation  of  the  sinhing  fund  of  a 
commercial  or  financial  undertaking  owing  to  the  conversion  of 
such  part  of  the  loan  into  ordinary  share  capital  or  stocJx  of  the 
undertaking  :  — 

in  ivhich  the  original  annual  instalment  was  found  by 
calculation  based  upon  a  specified  period  of  repayment  and 
rate  of  acctimulation.  Statonent  XVIII.  A. 


200    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

(1)  Ascertain,  in  the  manner  described  in  Chapter  XV^ 
whether  there  is  a  surplus  or  a  deficiency  in  the  fund 
apart  from  the  speciM  circumstances  now  und-er  reojiew, 
and  if  so,  calculate  the  corrective  annual  sinldng  fund 
instalment  required,  by  one  of  the  methods  there 
described.  Calculations  {X.V)  3  or  {X.VI)  1. 

(2)  Calculate    the    animal   sinking  fund   instalment,    whicli, 

if  set  aside  for  the  whole  of  the  unexpired  portion  of  the 
repayment  period,  icill  provide  the  part  of  the  loan 
converted  into  ordinary  share  capital,  and  thereby  with- 
drawn from  the  operation  of  the  fund. 

Calculation  (XVIII)  1. 

(3)  Deduct   the  annual   instalment  so   ascertained  from    the 

original  annual  instalment,  and  adjust  the  latter  if 
required,  by  the  above  corrective  instahnent. 

Calculation  {XYI)  1. 

(4)  The  remainder  icill  be  the  future  reduced  annual  iiistal- 

ment,  to  be  set  aside  and  accumulated  during  the  whole 
of  the  unexpired  portion  of  the  repayment  period. 

Calculation  (XVUI)  2. 

(0)  Prepare  a  statement  showing  the  final  repayment  of  the 

loan  by  the  operation  of  the  sinking  fund  under  the 
amended  conditions.  Statement  X]  III.  B. 

(6)  Prepare  a  pro  forma  account  showing  the  amount  which 
should  be  in  the  fund  at  the  end  of  ea-ch  year  of  the 
unexpired  repayment  period.     Pro  forma  Account,  .A  o.  6. 

Yariation  TV  (SuRPLi'S),  arising  on  the  withdrawal  of  part 
of  the  loan  from  the  operation  of  the  sinhing  fund  of  a 
commercial  or  financial  undertaking  oiring  to  the  conversion  of 
such  part  of  the  loan  into  ordinary  share  capital  or  stock  of  the 
undertaking  :  — 

in  wlvich  the  original  annual  instalment  is  a  stated  sum, 
and  is  7iot  based,  except  in  a  general  way,  upon  any  period 
of  repayment  or  rate  of  accumulation. 

Statement  XYIU.  D. 

(1)  Ascertain  from  the  actual  records  the  value  of  the  present 

investments  representing  the  fund.  Ascertain  also  the 
rate  of  income  yielded  on  such  value,  and  upon  this  and 
other  considerations,  as  elsewhere  described,  base  the 
future  rate  of  accumulation  of  the  fund. 


A    SURPLUS    IN    THE    FUND  201 

(2)  Ascertain   by  inspection  of  Table  III,    the   ayfroximate 

number  of  years  in  which  the  stated  annual  instalment 
ivill  accumulate  to  the  amount  of  the  origiiial  loan  at  the 
rate  of  accumulation  fixed  as  in  (1).  Adopt  the  nearest 
integral  number  of  years  so  found  as  the  approximate 
period  of  repayment  of  the  original  loan,  at  the  rate 
of  accumulation,  ascertained  as  above, 

(3)  Calculate  the  annual  sinking  fund  instalment  required  to 

repay  the  full  amount  of  the  loan  at  the  end  of  the 
approximate  period  of  repayment  found  in  {2)  at  the  rate 
of  accumidation  fixed  as  in  (-7). 

Calculation  (XVIII)  5.    £7441-63. 

(4)  Calculate  the  amount  which  would  be  in  the  fund  if  the 

annual  instalment  {£7 441' 63)  so  found  [3)  had  been  set 
aside  and  accitmulated  at  the  rate  per  cent,  fixed  in  (1) 
from  the  date  of  issue  of  the  loan  until  the  date  of 
con/version  of  part  of  the  loan. 

Calculation  (XVIII)  9.    £57021-21. 

(5)  Ascertain  the  apparent  surplus  or  deficiency  in  the  fund 

by  comparing  the  value  of  the  present  investments  repre- 
senting the  fund  (1)  with  the  amount  found  in  (4). 

Surplus,  £447-27. 

(6)  Calculate   the   corrective  instalment,    being   the   annuity 

ivhich  might  now  be  purchased  with  the  amount  found 
in  [5),  for  the  unexpired  portion  of  the  approximate 
repayment  period  (2)  at  the  rate  of  accumulation  (1). 

Calculation  (XVIII)  10.    £57-45. 

(7)  Calculate  the  annual  sinldng  fund  instalment  which,  if 

set  aside  for  the  unexpired  portion  of  the  approximate 
repayment  period  (2),  would  p>rovide  the  portion  of  the 
loan  converted  into  ordinary  share  cap<ital  or  stock. 

Calculation  (XVIII)  S.    £4429-52. 

[Here  refer  to  the  memo,  after  (12).'] 

(8)  Deduct     from     the     annual     sinking     fund     instalment 

(£7441-63)  found  as  in  (3),  but  not  from  the  fixed  instal- 
ment (£7,500)  originally  specified  in  the  trust  deed  and 
actually  set  aside,  the  annual  instalment  (£4429-52) 
found  in  (7). 


202  REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 

(9)  The  remainder  [£3012'11),  after  adjustment  in  resyect  of 
the  corrective  instalment  [£o7'45)  fo^l.nd  In  (6)  will  be 
the  future  reduced  annual  sinking  fund  instalment,  to  be 
set  aside  and  acciimulated  during  the  whole  of  the 
unexpired  portion  of  the  approximate  repayment  period 
found  in  {2).  '  £2954-66. 

(^10)  Prepare  a  statement  showing  the  final  repayment  of  the 
loan  by  the  operatio7i  of  the  fund  based  (not  tipon  the 
annual  instalment  (£7,500)  originally  fxed),  but  upon 
the  annual  instalment  (£7441'63)  found  in  (3),  as  subse- 
quently reduced  by  the  annual  instalment  (£4429'52) 
found  in  (7)  and  adjusted  by  the  annual  instalment 
{£57-45)  fotmd  in  (6).  Statement  XVIII.  D. 

(11)  If   the    anrmal   instalment    (£7441-63)    as    found    in    (3) 

differs  considerably  from  the  annual  instalment  (£7,500) 
originally  specified  in  the  trust  deed,  an  adjustment  may 
be  made  which  icill  have  the  effect  of  slightly  increasing 
or  reducing  the  annual  instalment  found  in  (9),  as  here- 
after described. 

(12)  Prepare  the  pro  forma  account  mentioned  in  the  previous 

methods.  Pro  forma  Account,  No.  6. 

Memo.  If  the  above  method  be  used  to  adjust  a  surplus 
in  the  fimd  arising  in  consequence  of  the  payment  into  the 
fund  of  the  proceeds  of  realisation  of  part  of  the  assets,  as 
described  in  Variation  II  (Surplus)  in  Chapter  X\  II,  biit  in 
whicJi  the  aiinual  instalment  is  a  stated  sum  not  found  by 
calcAilation,  substitute  operation  (2)  in  that  method  for  the  above 
operation   (7),  iiamely  :  — 

(2)  Calculate  the  annuity  which  may  be  purchased  for  the 
unexpired  portion  of  the  repayment  period  with  the 
amount  now  paid  into  the  fund.        Calculation  (XVII)  1. 


General  Eemarks  as  to  the  Sinking  Funds  of  Commercial 
AND  Financial  Undertakings. 

In  the  case  of  local  authorities  the  method  by  which  loans 
are  required  to  bo  repaid  by  means  of  a  sinking  fund  is  well 
defined,  but  in  the  case  of  commercial  or  financial  undertakings 
the  conditions  are  much  more  variable,  and  the  trust  deed  may 
stipulate  that  it  shall  be  provided  either  by  :  — 


A    SURPLUS    IN    THE    FUND  203 

(1)  An  equal  annual  instalment  to  be  calculated  on  tlie  basis 

of  a  given  repayment  period  and  a  prescribed  rate  of 
accumulation,  similar  to  those  of  local  authorities    or 

(2)  A  stated  sum  to  be  set  aside  each  year. 

Both  methods  must  be  considered.  In  order  to  attract  investors 
a  commercial  or  financial  undertaking,  when  inviting  subscrip- 
tions for  bonds,  debentures,  debenture  stock,  or  loan  capital  of 
any  other  nature,  may  give  the  investor  the  option  at  a  future 
date,  which  may  be  specified  or  not,  of  converting  the  loan  into 
share  capital  or  stock  of  the  undertaking,  on  what  may  then  be 
very  advantageous  terms  if  the  concern  be  making  good  profits. 
In  the  meantime  a  sinking  fund  is  required  by  the  trust  deed  to 
be  set  aside  out  of  profits  in  order  to  repay  the  total  loan  on  a 
given  date,  the  fund  to  accumulate  at  a  rate  per  cent.,  which 
may  be  specified  or  not,  by  investment  in  outside  securities. 

During  the  earlier  years,  if  profits  are  low,  the  provision  of 
the  annual  instalment  will  have  the  effect  of  reducing  the 
dividends  which  may  be  paid  to  the  ordinary  shareholders,  and 
there  will  not  therefore  be  any  inducement  to  the  loan  creditors 
to  give  up  their  security.  But  a  time  may  come  when  the 
position  of  the  undertaking  has  been  materially  improved,  and 
if  the  profits  have  been  good  and  are  likely  to  continue  so,  some 
of  the  loan  creditors  may  be  induced  to  convert  their  holding 
into  ordinary  share  capital  or  stock.  The  amount  of  loan  so 
converted  will,  of  course,  correspondingly  reduce  the  amount  to 
be  finally  provided  by  means  of  the  sinking  fund,  and,  seeing" 
that  the  period  of  repayment  of  the  balance  of  the  loan  will 
remain  unchanged,  the  effect  of  the  partial  conversion  will  be 
seen  solely  in  a  reduction  in  the  future  annual  instalment  to 
be  set  aside  out  of  profits  during  the  unexpired  portion  of  the 
original  repayment  period.  This  reduction  in  the  future  annual 
instalment  arises  in  consequence  of  two  factors,  namely,  the 
amount  of  loan  withdrawn  from  the  operation  of  the  fund  by 
reason  of  its  conversion  into  ordinary  share  capital  or  stock; 
and,  further,  from  the  fact  that  the  amount  now  in  the  fund 
represents  the  accumulation  of  past  instalments  set  aside  to 
provide  the  whole  of  the  loan.  Stated  in  terms  of  the  balance 
of  loan  still  unconverted,  there  is  a  present  surplus  in  the  fund 
due  to  setting  aside  in  the  past  Avhat  will  in  future  be  excessive 
annual  instalments,  and  there  is  also  an  excessive  future  annual 
instalment,  both  of  which  factors  have  been  dealt  with 
individually  in  previous  chapters.  They  are  here  combined: 
and  the  problem  is  further  complicated  by  the  nature  of  the 


204    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

animal  instalment.  If  it  be  a  stated  sum,  it  is  very  probable 
tliat  it  was  fixed  originally  AA-itli  only  an  approximate  regard  to 
the  period  of  repayment,  and  it  is  therefore  necessary  to 
ascertain  not  only  the  future  period  of  repayment  but  also  the 
future  rate  of  accumulation.  This  future  rate  of  accumulation 
may  be  based  upon  the  rate  of  income  now  yielded  by  the 
present  investments  of  the  fund,  and  therefrom  it  is  possible 
io  calculate  the  period  of  repayment,  both  of  which  are 
governing  factors  in  the  adjustment  to  be  made.  The  problem 
may  be  further  complicated  by  other  variations  in  the  period 
or  rate  per  cent.,  or  by  a  combination  of  both,  but  attention 
will  be  directed  only  to  the  above  factors. 

Taeiation  III  (Surplus),  arising  on  the  withdrawal  of  part  of 
the  loan  from  the  operation  of  the  sinking  fund  of  a 
commercial  or  financial  undertaking  owing  to  the  conversion 
of  such  part  of  the  loan  into  ordinary  share  capital  or  stock 
of  the   undertaking  :  — 

in  which  the  original  an/iual  instahneiif  wa.'i  found  by 
calculation,  based  upon  a  specified  period  of  repayment 
and  rate  of  acciiinulation. 

Statement  XVIII.  A. 

The  above  variation  will  be  illustrated  by  the  information 
previously  obtained  with  regard  to  the  imaginary  sinking  fund 
already  discussed,  namely,  the  same  amount  of  original  loan, 
£26,495,  repayment  period  25  years,  rate  of  accumulation  3^  per 
cent.,  amount  in  the  fund  £9,463,  or  a  deficiency  of  £469"T4. 
The  assumed  conversion  of  part  of  the  loan  takes  place  at  the 
end  of  the  12th  year  and  affects  £5,000  of  the  loan.  The 
■original  annual  instalment,  £680234  was  arrived  at  bv 
Calculation  (XY)  1. 

Statement  XVIII,  A,  following,  shows  the  successive  steps 
in  the  adjustment  of  the  annual  instalment,  and  Statement 
XVIII,  B,  shows  the  ultimate  repayment  of  the  loan  by  the 
■operation  of  the  fund  after  making  such  adjustment. 

The  above  deficiency  of  £469"T4  has  been  purposely 
introduced  into  this  example  in  order  to  demonstrate  that  the 
method  adopted  Avill  apply  to  a  combination  of  factors  requiring 
the  adjustment.  It  will  illustrate  the  remark  made  in  a 
previous  chapter  that  it  is  not  absolutely  necessary  to  ascertain 
the  exact  amount  of  the  deficiency  at  the  time  of  making  the 
adjustment  seeing  that  the  calculation  is  based  upon  the  actual 
amount  now  in  the  fund  and  the  acruniulation  thereof  at  the 


A    SURPLUS    IN    THE    FUND  205 

future  rate  per  ceut.  This  is  clearly  showu  by  the  examples- 
worked  out  iu  this  and  other  chapters  by  the  annual  increment 
(balance  of  loan)  method  in  which  no  mention  is  made,  or 
account  taken,  of  any  surplus  or  dehciency  in  the  amount  in 
the  fund  as  compared  with  the  amount  Avhich  should  be  in  the- 
fund  at  the  time  of  making  the  adjustment. 

The  reduction  of  £310'308  in  the  original  annual  instalment 
is  the  sole  effect  of  the  withdrawal  of  the  £5,000  of  loan  from 
the  operation  of  the  fund,  since  there  is  not  in  this  instance 
any  increase  in  the  income  to  be  added  to  the  fund,  as  found 
in  Chapter  XVII  was  the  elfect  of  the  payment  into  the  fund 
of  tbe  sum  of  £4,560  arising  out  of  the  proceeds  of  sale  of  part 
of  the  security  for  the  loan.     See  Statement  XYII.  A. 

This  method  of  adjusting  a  surplus  in  a  sinking  fund,  owing 
to  the  withdrawal  of  part  of  the  loan  from  the  operation  of  the 
fund,  should  therefore  be  carefully  compared  with  the  method 
found  necessary  in  the  case  of  a  surplus  arising  from  a  cash 
payment  into  the  fund  from  proceeds  of  assets  realised.  This- 
will  be  fully  considered  at  the  end  of  this  chapter. 

The  Annual  Inckemext  (ualaxce  of  loan)  Method.  The 
method  of  arriving;  at  the  amended  annual  instalment  based 
upon  the  future  annual  increment  is  summarised  at  the 
beginning  of  Chapter  XV,  and  is  fully  described  in  Chapter 
XVI.  As  the  method  about  to  be  discussed  is  based  upon  the 
same  conditions  as  in  the  previous  example,  Statement  X^  III. 
B.,  showing  the  final  repayment  of  the  loan  will  again  apply. 
This  method  is  shown  in  Statement  XVIII.  C  following. 


2o6    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


A  Surplus  in  the  Fund.  Statement  XVIIL  A. 

The  Deductive  Method. 

Showing  the  method  of  adjusting  the  annual  instalment  in 
consequence  of  a  surplus  in  the  fund,  arising  on  the  with- 
drawal of  part  of  the  loan  from  the  operation  of  the  fund 
owing  to  the  conversion  of  such  part  of  the  loan  into 
ordinary  share  capital  or  stock. 

Yariatiox  III  (SuEPLus),  in  which  the  original  annual  instal- 
ment was  found  bj'  calculation,  based  upon  a  specified 
period  of  repayment  and  rate  of  accumulation. 

Calculation  (XY)  1. 

Equivalent 
Annual  amount 

instalment.       of  original  loan. 

Amount  of  originalloan £26495  00 

Original  annual  instalment. 

Calculation  (XY)  1  £680-234 

Additional  annual  instalment  to  provide 

present    deficiency   in   the   fund. 

Calculation  (XYj  3     £45-594 


£725-828 


Present  investments  (at  end  of  12th  year) 

£9463-00 


Amount  thereof,  accumulated  for  13 
years  at  3^  per  cent. 

Calculation  (XY)  4  £14799-71 


£725-828  £11695-29 


Loan  withdrawn  from  the  operation  of 

the  fund  at  the  end  of  the  12th  year  £5000-00 

equivalent    to    a    reduction    in    the 
annual  instalment  of 

Calculation  (XYIII)  1  £310308 


£415-520     £6695-29 


Amended  annual  instalment  of  £415-520 
will  provide  £6695-29  in  13  years 
at3l  per  cent.    Calculation  (XYIII)  2  £415520     £6695-29 


A    SURPLUS    IN    THE    FUND  207 


A  Surplus  in  the  Fund.  Statement  XVIII.  B. 

Showing  the  final  repayment  of  the  loan,  by  the  operation  of 
the  sinking  fund  after  making  the  adjustment  in  the  annual 
instalment  consequent  upon  a  surplus  in  the  fund,  arising 
on  the  withdrawal  of  part  of  the  loan  from  the  operation 
of  the  fund  owing  to  the  conversion  of  such  part  of  the  loan 
into  ordinary  share  capital  or  stock. 

Variation  III  (Surplus),  in  which  the  original  annual  instal- 
ment was  found  by  calculation,  based  upon  a  specified 
period  of  repayment  and  rate  of  accumulation. 

Calculation  (XV)  1. 


Present  investments  (at  end  of  12th  year) 

£9463-00 


Equivalent 
Annual  amount 

instalment         of  original  loan. 


Amount  thereof,  accumulated  for  13 
years  at  3^  per  cent. 

Calculation  (XV)  4  £14799-71 

Amended  annual  instalment : — 

Original  annual  instalment     £680'234 

Additional  annual  instalment  to 
provide  the  present  deficiency  of 
£469-744.         Calculation  (XVI)  1     £45-594 


£725-828 


Reduced   annual   instalment   due  to 
withdrawal  of  £5,000  of  loan. 

Calculation  (XVIII)  1  £310-308 


£415-520 


Amount  thereof,   in  13  years  at  3^  per  cent. 

Calculation  (XVIII)  2     £6695-29 


Amount  in  the  fund  at  the  end  of  25  years £2149500 


heing  amount  of  original  loan       ...  £26495-00 
less  the  amount  converted  as  above     £5000- 00 


£21495-00 


2oS  REPAYMENT    OF    LOCAL   AND    OTHER    LOANS 


A  Surplus  in  the  Fund.  Statement  XVIII.  C. 

The  Annual  Increment  (balance  of  loan  )  Method. 

To  find  the  amended  annual  sinking  fund  instalment  consequent 
upon  a  surplus  in  the  fund,  arising  on  the  withdrawal  of 
part  of  the  loan  from  the  operation  of  the  fund,  owing  to 
the  conversion  of  such  part  of  the  loan  into  ordinary  share 
capital  or  stock. 

Variation  III  (Surplus),  in  which  the  original  annual  instal- 
ment was  found  by  calculation,  based  upon  a  specified 
period  of  repayment  and  rate  of  accumulation. 

Calculation  (XV)  1. 

Amount  of  original  loan  (25  years)  .. . £2649500 

deduct  portion  thereof  converted  into  ordinary 
share  capital  or  stock  and  withdrawn 
from  the  operation  of  the  fund  at  the 
end  of  the  12th  year £5000-00 


£21495-00 


deduct  amount  in  the  fund  at  the  end  of  the 

12th   vear      £9463-00 


Balance  of  loan    £1203200 


Amended  annual  increment  to  be  added  to  the 
fund,  and  accumulated  at  3-|  per  cent,  to 
provide  this  amount  at  the  end   of   13   years 

Calculation  (XVIII)  3       £746-725 
deduct  income  to  be  received  from  the  present 

investments  (£9,463)  at  3^  per  cent.       £331-205 


Amended  annual  instalment,  being      £415-520 

Original   annual   instalment    £680234 

increased  by       £45-594 


£725-828 
and  reduced  by £310-308 


£415-520 


The  rule  relating  to  tliis  method  is  stated   at  tlie  head   of 
Chapter  XXII. 


A    SURPLUS    IN    THE    FUND 


209 


Pro  forma  Sinking  Fund  Account,  No.  5. 

A  Surplus  in  the  Fund.     (Variation  III.) 

Loan  of  £26,495,  repayable  at  the  end  of  25  years. 

Showing  the  final  repayment  of  the  loan,  by  the  operation  of 
the  reduced  annual  instalment  of  £415'520. 


Statement  XVIII.  B. 


Hate  of  accumulation,  3|  per  cent. 


Year. 

Amount  in 

the  fund  at 

beginning  of 

year. 

Income 
received  from 
investments 
3J  per  cent. 

Annual 
sinking  fund 
instalment. 

Amount  in 
tlie  fund 
at  end 
of  year. 

Year. 

1 

1 

2 

2 

3 

3 

4 

The  amount  in  the  f 

und  at  the  ( 

?nd  of  the 

4 

5 

12th  year,  £9,463,  is 

1  an  assume! 

d  amount, 

5 

6 

and    is 

equivalent 

to    setting 

aside    an 

6 

7 

annual 

instalment  of  £648-064, 

as  shown 

7 

8 

by  Calculation  (XVI)  10,  instead  of  the 

8 

9 

correct  annual  insta 

Iment  of  £680-234. 

9 

10 

10 

11 

11 

12 

9463-000 

12 

13 

9463-000 

331-205 

415-520 

10209-725 

13 

14 

10209-725 

357-340 

415-520 

10982-585 

14 

15 

10982-585 

384-390 

435-520 

11782-495 

15 

16 

11782-495 

412-387 

415-520 

12610-402 

16 

17 

12610-402 

441-364 

415-520 

13467-286 

17 

18 

13467-286 

471-355 

415-520 

14354-161 

18 

19 

14354161 

502-396 

415-520 

15272-077 

19 

20 

15272077 

534-523 

415-520 

16222-120 

20 

21 

16222120 

567-774 

415-520 

17205-414 

21 

22 

17205-414 

602-189 

415-520 

18223123 

22 

23 

18223-123 

637-809 

415-520 

19276-452 

23 

24 

19276-452 

674-676 

415-520 

20366-648 

24 

25 

20366-648 

712-832 

415-520 

21495000 

25 

210    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Variation  IY.  (SuEPLrs),  arising  on  the  withdrawal  of  part  of 
the  loan  from  the  operation  of  the  sinking  fund  of  a 
commercial  or  financial  undertaking  owing  to  the  con- 
version of  such  part  of  the  loan  into  ordinary  share  capital 
or  stock  of  the  undertaking  :  — ■ 

in  ivhicli  the  original  annual  instalment  is  a  stated  sum 
and  is  not  based  except  in  a  general  way  upon  any 
period  of  repayment  or  rate  of  accumulation. 

Statement  XVIII.  D. 

This  variation  is  similar  in  principle  to  the  one  last 
discussed,  but  requires  different  treatment  owing  to  the  fact 
that  the  original  annual  instalment  is  a  stated  sum  arrived  at 
in  a  somewhat  empirical  manner,  without  any  calculation 
similar  to  (XV)  1,  which  is  based  upon  a  prescribed  period  of 
repayment  and  rate  of  accumulation.  In  the  following 
example,  used  to  illustrate  the  variation  under  review  new  data 
have  been  adopted,  and  the  question  is  dealt  with  solely  as 
regards  the  loan  debt  of  commercial  or  financial  undertakings 
without  making  any  comparison  with  the  methods  to  be  adopted 
in  the  case  of  a  local  authority.  In  the  early  days  of  municipal 
finance  the  annual  instalment  to  be  set  aside  was  often  a  fixed 
amount,  being  generally  a  definite  percentage  of  the  amount 
of  the  loan,  but  the  conditions  then  imposed  upon  such 
authorities  were  vague  and  indefinite  both  as  regards  the 
accumulation  of  the  fund  and  the  gradual  repayment  of  the  debt, 
and  left  entirely  out  of  account  the  life  or  duration  of  utility 
of  the  asset  created  out  of  the  loan. 

Example  to  Illustrate  Variation  IV  (Surplus).  The 
sinking  fund  under  review  relates  to  the  repayment  of  a  loan 
of  £150,000,  and  an  annual  instalment  of  £7,500  is  required 
to  be  set  aside  for  this  purpose  out  of  the  profits  of  the  undertak- 
ing and  invested  in  outside  securities.  Under  the  trust  deed  the 
loan  creditors  have  the  option  of  converting  their  holding  into 
ordinary  share  capital  or  stock  at  any  time  within  seven  years 
from  the  date  of  issue.  The  price  payable,  on  conversion,  for 
the  ordinary  share  capital  is  immaterial  for  the  present  purpose, 
as  is  also  the  rate  of  interest  payable  upon  the  loan;  but  any 
premium  payable  to  the  loan-holders  upon  conversion  or 
redemption  should  be  taken  into  account.  At  the  end  of  the 
seventh  year  the  holders  of  £45,000  of  loan  elect  to  exercise 
the  above  option,  and  convert  their  loan  holding  into  ordinary 
share   capital  or  stock.      Seeing   that   no   specified    period    is 


A    SURPLUvS    IN    THE    FUND  211 

prescribed  within  which  the  loan  shall  be  repaid  by  means  of 
the  sinking  fund  and  that  the  annnal  sum  to  be  set  aside  is 
fixed  at  £7,500,  there  was  not  any  necessity,  at  the  date  of  issue 
of  the  loan,  to  make  any  calculation  of  the  annual  instalment 
as  in  the  case  of  the  sinking  funds  of  local  authorities.  This 
annual  instalment  of  £7,500,  it  will  be  assumed,  has  been 
regularly  set  aside  and  invested  each  year,  and  at  the  end  of 
the  seventh  year,  when  £45,000  of  original  loan  is  converted 
into  ordinary  share  capital  or  stock,  it  will  have  amounted  to 
£5746848,  having  earned  an  average  accumulation  rate  of 
8  per  cent.,  as  shown  by  Calculation  (XYIII)  4.  In  actual 
practice,  of  course,  this  amount  would  be  obtained  from  the 
actual  records  or  books  of  account. 

The  position  at  the  end  of  the  seventh  year  will  therefore  be 
as  follows  :  — 

1.  Loan  outstanding  and  unconverted       £105,000 


2.  Amount  in  the  sinking  fund,   invested  and 

yielding  o  per  cent,  per  annum £57468'48 


3.  Present  annual  instalment     £7500* 


This  annual  instalment  will  be  reduced  in  future  years  owing 
to  the  withdrawal  of  £45,000  of  loan  from  the  operation  of  the 
fund. 

The  next  step  in  the  adjustment  is  to  ascertain  the  annual 
amount  by  which  this  instalment  may  be  reduced  and  yet  fulfil 
the  original  obligation  to  repay  the  unconverted  portion  of  the 
loan  under  the  original  conditions.  There  are  several  ways  of 
doing  this,  as  may  be  gathered  from  previous  examples.  But 
it  is  in  any  case  first  essential  that  the  estimated  future  rate  of 
accumulation  shall  be  fixed.  In  this  case  past  experience  is 
available,  and,  for  convenience,  3  per  cent,  will  be  taken,  being 
the  rate  of  income  already  yielded  by  the  present  investments 
representing  the  fund.  Any  variation  in  this  rate  per  cent, 
and  in  the  future  accumulation  rate  may  be  treated  as 
explained  in  Chapter  XXI  (variation  in  the  rates  per  cent.). 
Having  decided  upon  the  future  estimated  rate  of  accumulation, 
it  is  next  necessary  to  fix  the  period  of  redemption  in  order  to 
ascertain  the  reduction  in  the  annual  instalment  of  £7,500. 
This  would  not  be  necessary  but  for  the  amount  at  present  in 
the  fund.  The  instalment  then  would  be  lo^^^^^ths  of  the 
original  instalment  of  £7,500,  or  £5,250  per  annum,  but  this 


212    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

will  be  reduced  by  au  annual  amount  depending  upon  tlie 
money  now  in  tlie  fund.  There  are  therefore  two  factors  to  be 
taken  into  account — (1)  the  accumulation  of  the  £57468'48 
now  in  the  fund,  and  (2)  the  accumulation  of  the  future 
reduced  annual  instalment  which  it  is  required  to  ascertain. 
It  is  not  possible  to  combine,  in  one  calculation,  factors 
involving  the  amount  of  £1,  and  also  of  £1  per  annum 
without  reducing  both  to  a  common  denomination,  and  there- 
fore it  is  better  as  the  simplest  method  to  revert  to  the  original 
conditions  at  the  date  of  the  issue  of  the  loan. 

The  first  step  is  to  ascertain  the  approximate  number  of 
years  in  which  an  annual  instalment  of  £7,500  will  amount  to, 
and  repay,  a  loan  of,  £150,000  if  accumulated  at  -3  per  cent. 
per  annum.  This  is  the  rate  of  income  which  has  been  yielded 
by  the  present  investments  representing  the  fund  and  which  it 
is  assumed  will  continue  to  be  yielded  by  any  future  invest- 
ments. The  number  of  years  may  be  ascertained  approximately 
by  an  inspection  of  Table  III,  seeing  that  if  £7,500  per  annum 
will,  at  3  per  cent.,  amount  to  £150,000,  £1  per  annum  will,  at 
the  same  rate  and  in  the  same  time,  amount  to  £20.  Table  III 
gives  the  following  figures  :  — 

£1  per  annum  will,  at  3  per  cent.,  amount  to — 

in  15  years £1859891, 

in  16  years 20-15688, 

in  17  years 21"76159, 

and  an  even  16  years  is  therefore  adopted  as  the  approximate 
period  of  repayment,  which  will  be  slightly  in  excess  of  the 
actual  period  required,  and  the  calculated  annual  instalment 
will  therefore  be  less  than  £7,500.  In  order  to  make  the 
calculation  in  such  a  manner  that  the  result  may  be  proved  as 
in  other  cases,  it  is  necessary  to  first  ascertain  the  exact  annual 
instalment,  to  be  accumulated  at  3  per  cent.,  to  repay  £150,000 
in  exactly  16  years.  The  annual  instalment  is  £7441"63,  as 
shown  by  Calculation  (XVIII)  5. 

This  annual  instalment  of  £7441-63  is  less  than  the  stated 
annual  instalment  of  £7,500,  as  will  be  gathered  from  the  above 
extracts  from  Table  III,  which  show  that  £1  per  annum  will 
in  16  years,  at  3  per  cent.,  amount  to  £2015688;  consequently 
the  prescribed  annual  instalment  of  £7,500  will  amount  to  :  — 

(£20-15688  X  £7,500)  or  to  £151,17659 

in  16  years  at  3  per  cent.,  as  shown  by  Calculation  (XYIII)  6. 


A    SURPLUS    IN    THE    FUND  213 

Expressed  in  terms  of  the  above  difference  it  will  be  seen  tliat : 

£7,500 -£7441-63  per  annum  or  £58-37 

will  in  16  years,  at  3  per  cent.,  amount  to 

£151,176-59 -£150,000  or        £1176-58 

as  shown  by  Calculation  (XYIII)  7. 

By  adopting  the  above  annual  instalment  of  £7441-63 
instead  of  £7,500,  an  intentional  error  of  £58-37  per  annum  is 
introduced,  relating-  to  the  repayment  of  a  loan  of  £150,000  in 
16  years.  But  the  reduced  annual  instalment  which  is  required, 
and  which  will  be  based  upon  the  above  annual  instalment  of 
£7441-63,  will  relate  to  a  loan  of  £105,000  repayable  in  9  years 
only.  This  intentional  error  may  be  corrected  if  thought 
desirable  or  required  in  the  manner  to  be  afterwards  explained. 
The  following  data  have  now  been  ascertained: —  A  loan 
of  £105,000  is  repayable  in  a  period  of  9  years,  and  towards 
this  there  is  in  the  fund  an  amount  of  £57468-48,  which,  it  is 
estimated,  will  accumulate  at  3  per  cent.  There  is  an  annual 
instalment  of  £7441-63  to  be  set  aside  for  9  years,  and  an 
intentional  error  of  £58-37  per  annum  has  been  introduced  into 
the  problem. 

It  is  required  to  find  the  annual  amount  by  which  the  above 
instalment  of  £7441-63  may  be  reduced,  consequent  upon  the 
withdrawal  of  £45,000  of  loan  from  the  operation  of  the  fund. 
The  problem  differs  somewhat  from  the  surplus  of  £4,560, 
already  considered  in  Chapter  XVII  (Statement  XVII.  A.). 
In  that  case  the  £4,560  was  paid  into  the  fund  in  consequence 
of  the  realisation  of  assets  forming  part  of  the  security  for  the 
loan,  and  was  applied  in  repaying  part  of  the  loan  or  remained 
to  swell  the  assets  of  the  fund. 

In  the  present  instance  the  conversion  of  £45,000  of  loan 
into  share  capital  or  stock  may  be  looked  upon  as  an  entirely 
separate  transaction,  and  may  be  regarded  as  so  much  cash 
received  in  consequence  of  the  issue  of  new  share  capital,  and 
applied  in  reduction  of  the  loan  debt.  The  undertaking, 
except  as  afterwards  mentioned,  does  not  derive  any  benefit 
from  the  substitution  of  its  obligation  to  the  new  shareholders 
for  its  obligation  to  the  previous  loanholders.  Indeed,  it  may 
happen  that  the  inducement  to  the  loanholders  to  convert  their 
secured  debt  into  ordinary  share  capital  or  stock  is  the  expecta- 
tion of  a  higher  rate  of  interest  upon  their  investment.  The 
only  benefit  to  the  undertaking  is  that  the  capital  is  firmly 
invested  in  the  concern;  and  the  saving  in  the  sinking  fund 
instalment,  if  previously  taken  out  of  profits,  will  help  to 
provide  any  increased  return  payable  by  way  of  dividend  to  the 


214    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

origiual  ioaniioldeis  iu  respect  of  the  loau  capital  so  converted. 
It  IS  here  necessary  to  depart  from  the  method  adopted  in 
dealing  with  the  sum  of  £4,560  paid  'into  the  fund  in  the 
example  considered  in  Chapter  XVII.  The  £4,560  was  an 
amount  actually  in  hand,  and  Calculation  (XVII)  1  shows  the 
method  of  finding-  the  future  annuity  which  it  would  purchase. 
The  £45,000  in  tlie  present  case,  on  the  contrary,  is  an  amount 
due  at  a  future  time,  namely,  at  the  end  of  the  sinking  fund 
period,  and  the  problem  therefore  becomes  inverted,  and  instead 
of  calculating  the  annuity  which  £45,000  will  purchase,  it  is 
required  to  ascertain  the  annuity  or  annual  sinking  fund 
instalment  which  in  the  remaining  unexpired  period  of  9  years 
will  amount  to  that  sum.  This  is  shown  by  Calculation 
(XVIII)  8,  which  may  be  usefully  compared  with  Calculation 
(XVII)  1.  The  annual  instalment  so  found  is  £4429-52,  by 
which  amount  the  original  annual  instalment  of  £7441'63  may 
be  reduced,  making  the  amended  annual  instalment  £301211. 

There  is,  however,  a  further  slight  correction  to  be  made. 
In  order  to  simplify  the  calculation  an  even  period  of  16  years 
has  been  adopted,  which  is  in  excess  of  the  actual  period  of 
repayment  and  requires  a  reduced  annual  instalment  of 
£7441-63  instead  of  £7,500.  Until  the  conversion  of  part  of 
the  loan  into  ordinary  share  capital  or  stock,  the  undertaking 
had  been  setting  aside  an  annual  instalment  of  £7,500,  so  that 
there  is  now  an  apparent  surplus  in  the  amount  in  the  fund  as 
compared  with  what  would  have  been  in  the  fund  if  £7441-63 
only  had  been  annually  set  aside.  To  ascertain  the  amount  of  this 
surplus  it  is  requisite  to  ascertain  the  amount  to  which  an  annual 
instalment  of  £7441-63  will  accumulate  in  7  years  at  3  percent. 

This,  as  shown  by  Calculation  (XVIII )  9,  is £5702121 

and    on    comparing    this    sum    with    the    amount 

actually  in  the  fund,  being  the  accumulation 

of  the  stated  instalment  of  £7,500 

Calculation  (XVIII)  4,  viz.  :      £57468-48 

the  apparent  present  surplus  is  found  to  be         £447*27 

whioli  amouiii,  being  now  in  the  fund,  will  accumulate  for 
9  years  at  3  per  cent.,  and  is  the  present  value  of  an  annuity, 
as'shown  by  Calculation  (XVIII)  10,  of  £574446  which  may  be 
applied  in  further  reduction  of  the  annual  instalment  of 
£7441-63  in  the  same  way  that  the  annual  instalment  of 
£442001  was  applied  in  the  case  of  the  surplus  of  £4,560  in 
Chapter  XVII,  Statement  XVII,  A. 


A    SURPLUS    IN    THE    FUND  215 

In  the  event  of  the  calculated  insstalnient  found  as  above 
exceeding  the  prescribed  instalment  of  £7,500  there  would  be 
an  apparent  deficiency  in  the  fund,  instead  of  a  surplus,  which 
would  alter  the  method,  but  not  the  principle,  of  the  minor 
adjustment  under  consideration. 

The  A'arious  stages  of  the  adjustment  have  already  been  so 
fully  described  that  it  is  not  requisite  to  prepare  a  statement 
similar  to  XYIII,  A,  in  the  j)i"evious  example. 

A  statement  has,  however,  been  prepared,  similar  to 
XVIII,  B,  showing  the  final  repayment  of  the  loan  by  the 
operation  of  the  fund  after  making  the  above  adjustment  in  the 
annual  instalment.  See  Statement  XVIII,  D,  and  the  pro 
forma  account  No.  6  following. 

Correction  of  the  Intentional  Error.  There  now 
only  remains  the  correction  of  the  above  intentional  error  of 
£58'37  in  taking  the  annual  instalment  at  the  calculated 
amount  of  £7441'63  instead  of  £7,500,  which  lengthened  the 
period  of  repayment  by  part  of  a  year. 

The  £7,500,  or  any  other  similarly  prescribed  annual 
instalment,  is  generally  fixed  in  an  empirical  manner  with  only 
a  rough  approximation  to  the  actual  requirements  based  upon 
the  conditions  in  each  case.  It  may  therefore  be  concluded 
that  the  instalment  ascertained  in  the  above  manner  will  meet 
any  practical  need  likely  to  arise  in  such  a  case.  If,  however, 
there  is  at  any  time  a  necessity  for  greater  accuracy  it  may  be 
ascertained,  approximately,  by  the  following  method  : — • 
The  intentional  annual  error  introduced  was £58'37 


This  caused  an  apparent  surplus  in  7  years  of £447'27 


Equal  to  an  annual  instalment  spread  over  9  years  of       £57'45 


This  error  related  to  a  loan  of £150,000 


The  correction  will  relate  to  a  loan  of  only £105,000 


Therefore  the  correction  may  be  taken  as  ^''^/isoths 

of  £57-45.  or £40-215 


which  would  in  9  years  amount,  at  3  per  cent.,  to 

Calculation  (XVIII)  13     £408-549 


The  present  value  of  this  annual  sum  in  9  years,  at 

3  per  cent.,  is  Calculation  (XVIII)  14     £313-118 


2i6    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

aud  the  currectiou  may  be  made  by  increasing  each  of  the 
reduced  annual  instalments  of  £2954' 66  by  £40"215,  or  by 
paying  into  the  fund  at  the  present  time  the  above  present 
value  thereof,  namely,  £313"118.  If  no  such  correction  be 
made,  only  £408'549  of  original  loan  will  remain  unprovided  for 
at  the  end  of  the  period  (which  is  slightly  less  than  16  years)  in 
which  the  original  instalment  of  £7,500  would  have  repaid 
the  loan. 

The  Axxual  Inckemext  (balance  of  loan)  Method.  The 
method  of  arriving  at  the  amended  annual  instalment  based 
upon  the  future  annual  increment  is  summarised  at  the 
beginning  of  Chapter  XY,  and  is  fully  described  in  Chapter 
XVI.  As  the  method  about  to  be  discussed  is  based  upon  the 
same  data  as  in  the  example  previously  used,  the  following 
Statement  XYIII,  D,  showing  the  final  repayment  of  the  loan 
will  still  apply.  For  the  reasons  already  given  the  calculation 
cannot  be  made  in  terms  of  the  stated  instalment,  but  must 
be  made  in  terms  of  the  approximate  amount  of  £T441'63  found 
by  Calculation  (XYIII)  5.  This  method  is  shown  in 
Statement  XYIII,  E,  following. 


A  Surplus  in  the  Fund.  Statement  XVIII,  D. 

Showing  the  final  repayment  of  the  loan,  by  the  operation  of 
the  sinking  fund,  after  making  the  adjustment  in  the 
annual  instalment,  consequent  upon  a  surplus  in  the  fund, 
arising  on  the  withdrawal  of  part  of  the  loan  from  the 
operation  of  the  fund  owing  to  the  conversion  of  such  part 
of  the  loan  into  ordinary  share  capital  or  stock. 

Yariation  IY  (Surplus),  in  which  the  original  annual  instal- 
ment is  a  stated  sum,  and  is  not  based,  except  in  a  general 
way,  upon  any  period  of  repayment  or  rate  of  accumulation. 

Equivalent 
Annual  amount  of 

instalment.  original  loan. 

Present  investments  fat  end  of  7  years), 
Calculation    (XVIII)  4  £57468-48 


Amount    thereof,    accumulated    for 
9  years  at  3  per  cent. 

Calculation  (XYIII)  11  £74983-30 


A    SURPLUS    IN    THE    FUND  217 

Amended  annual  instalment  ;— 

Urigiual      annual      instalment,      as 

provided  by  trust  deed  ...      ^''''^OQ^O 


Substituted  annual  instalment  as 
adopted  in  Calculation  (XVIII)  5, 
based  upon  a  rate  of  accumulation 
of  3  per  cent,  and  a  repayment 
period  of  16  years £7441"63 

This  will  be  reduced  by  the  annual 
instalment  required  to  repay  the 
£45,000  of  loan  withdrawn,  in  9 
years  at  3  per  cent. 

Calculation  (XVIIIj  8  £442952 

£301211 

And  will  be  further  reduced  by  the 
annual  instalment  to  proAade 
£447-27,  being  the  surplus  which 
will  be  in  the  fund  at  the  end  of 
16  years,  due  to  taking  an  even 
period   of   16  years 

Calculation  (XVIII)  10       £57-45 

£2954-66 


Amount    thereof,     accumulated    for 
9  years  at  3  per  cent. 

Calculation  (XVIII)  12  £3001670 


Amount  in  the  fund,  at  the  end  of  16  years 

hemg  the  original  loan       £15000000 

reduced  by  the  amount  of 
loan  withdrawn  from  the 
operation  of  the  fund  ...     £4500000 


£10500000 


£10500000 


This  statement  shows  the  method  of  making  the  correction 
in  the  annual  instalment,  and  also  the  final  repayment  of  the 
loan.  The  amended  annual  instalment  may  also  be  found  by 
the  annual  increment  (balance  of  loan)  method,  as  shown 
in  Statement  XVIII.  E. 


2lS 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


A  Surplus  in  the  Fund. 


Statement  XVIII.  E. 


The  Annual  Increment  (balance  of  loan)    Method. 

To  find  the  ameuded  auimal  sinking-  fund  instalment,  couse- 
qnent  upon  a  surplus  in  the  fund,  arising  on  the  withdrawal 
of  part  of  the  loan  from  the  operation  of  the  fund  owing 
to  the  conversion  of  such  part  of  the  loan  into  ordinary 
share  capital  or  stock. 

Variation  IV  (Surplus),  in  which  the  original  annual  instal- 
ment is  a  stated  sum,  and  is  not  based,  except  in  a  general 
way,  upon  any  period  of  repayment  or  rate  of  accumulation. 

Amount  of  original  loan  (16  years) £15000000 

deduct,  portion  thereof  converted  into  ordinary 
share  capital  and  withdrawn  from  the 
operation  of  the  fund  at  the  end  of 
the  7th  year £45000-00 


£10500000 
deduct  amount  in  tlie  fund  at  the  end  of  the 

7th    year       £57468-48 


Balance  of  loan 


...     £475:U-52 


Amended  annual  increment  to  be  added  to  the 
fund,  and  accumulated  at  3  per  cent,  to  provide 
this  amount  at  the  end  of  9  years 

Calculation  (XVIII)  15 

deduct  income  to  be  received  from  the  present 
investments  (£57468-48)  at  3  per  cent. 


£4678-71 


£172405 


Amended  annual  instalment,  being  : — 

Calculated  instalment       

reduced  by       •• 


£7441-63 

£4480-97 


£2954-66 


£295406 


The  final  repayment  of  tlie  h)an  by  tlie  operation  of  the 
sinking  fund,  after  making  the  above  adjustment  in  the  annual 
instalment  is  shown  in  Statement  XVIII.  D. 


A    SURPLUS    IN    THE    FUND 


219 


Pro  forma  Sinking  Fund  Account,  No.  6, 

A  Surplus  in  the  Fund.     (Variation  IV.) 

Loan  of  Hr5(),()l)0,   repayahJc  bi/  a  statcil  annual  insfahncnf  of 

£7,600. 

Showing  the  final  repayment  of  tlie  balance  of  unconverted 
loan,  by  the  operation  of  the  reduced  annual  instalment  of 
£2954-660. 


Statement  XVIII.  D. 


Rate  of  accumulation,  3  per  cent. 


Year. 

Amount  in 

the  fund 

at  beginning 

of  year. 

Income 
received  from 
investments 
3  per  cent. 

Annual 
sinking  fund 
instalment. 

Amount  in 
tlie  fund 
at  end 
of  year. 

Year. 

1 

Nil 

Nil 

7500  000 

7500000 

1 

2 

7500000 

225000 

7500000 

15225-000 

2 

3 

15525000 

456-750 

7500000 

23181-750 

3 

4 

23181-750 

695-453 

7500000 

31377-203 

4 

5 

31377-203 

941-316 

7500-000 

39818-519 

5 

6 

39818-519 

1194-556 

7500()00 

48513-075 

6 

7 

48513-075 

1455-495 

7500000 

57468-480 

8 

57468-480 

1724-054 

2954-660 

62147-194 

8 

9 

62147-194 

1864-416 

2954-660 

66966-270 

9 

10 

66966-270 

2008-988 

2954-660 

71929  918 

10 

11 

71929-918 

2157-898 

2954  660 

77042-476 

11 

12 

77042-476 

2311-274 

2954-660 

82308-410 

12 

13 

82308-410 

2469-252 

2954-660 

87732-322 

13 

14 

87732-322 

2631-970 

2954-660 

93318-952 

14 

15 

93318-952 

2799-569 

2954-660 

99073181 

15 

16 

99073-181 

J. 

2972-159 
\.mount  of  loan 

2954-660 
converted 

105000000 

16 

45000000 

Comparison  of  Methods  Previously  Discussed.  This 
concludes  the  examination  of  the  various  methods  of  adjusting 
a  sinking  fund  to  compensate  for  a  difference  between  the  actual 
amount  in  the  fund  and  the  amount  which  should  be  in  the 
fund  at  any  time  in  order  to  carry  out  the  original  obligation ; 
and  the  results  may  be  summarised  as  follows. 


220    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

In  the  case  of  the  adjustment  (Yariation  I)  caused  by  a 
deficiency  in  the  fund,  as  shown  by  Statement  XYI,  A,  in 
Chapter  XYI,  the  deficiency  was  corrected  by  an  additional 
sinking  fund  instalment  to  be  set  aside  during  the  whole  of  the 
unexpired  portion  of  the  repayment  period,  as  shown  by 
Calculation  (XYI)  1. 

In  Yariation  II,  described  in  Chapter  XYII,  Statement 
XYII,  A,  relating  to  a  surplus  of  £4,560,  being  the  proceeds 
of  sale  of  assets,  paid  into  the  fund,  a  different  method  was 
adopted.  In  that  case  there  was  an  actual  increase  in  the  cash 
assets  of  the  fund  which  operated  in  two  ways,  (1)  by  increasing 
the  future  income  of  the  fund,  in  consequence  of  which  the 
present  sum  of  £4,560  will  ultimately  repay,  by  accumulation, 
£7131"64  of  original  loan,  and  (2)  by  reducing  the  amount  of 
loan  ultimately  repayable  by  £7131"64,  it  relieved  the  future 
years  of  the  amount  of  the  sinking  fund  instalment  (£442'60) 
equivalent  to  that  amount  for  the  unexpired  portion  of  the 
repayment  period  of  13  years,  which  is  the  annuity  which 
might  now  be  purchased  with  the  sum  of  £4,560.  In  the  two 
Yariations  III  and  lY  which  have  just  been  considered  there  is 
a  surplus  in  the  fund  caused  by  the  withdrawal  of  part  of  the 
loan  from  the  operation  of  the  fund.  There  is  here  no  actual 
addition  to  the  assets  of  the  fund,  as  in  the  case  of  the  payment 
into  the  fund  of  the  proceeds  of  sale  of  part  of  the  security  for 
the  loan.  The  surplus  may  in  effect  be  considered  as  a 
lightening  of  the  burden  previously  borne  by  the  undertaking 
measured  by  the  reduction  in  the  amount  of  loan  to  be 
ultimately  provided.  Consequently  the  surplus  operates  in  one 
direction  only,  namely,  by  reducing  the  original  annual  instal- 
ment to  be  set  aside,  whether  that  instalment  was  arrived  at 
by  calculation  in  the  ordinary  manner  or  was  a  round  sum 
specified  in  the  trust  or  other  deed  under  which  the  fund  was 
instituted. 

Further  Problems.  There  are  other  problems  which  may 
arise  in  connection  witli  the  sinking  funds  of  commercial  or 
financial  undertakings,  but  which  have  not  been  treated  in  an 
exhaustive  manner,  because  they  may  be  solved  by  one  or  other 
of  the  methods  elsewhere  described.     Thev  are  as  follows  :■ — 


Redemption  uy  Drawings  : 

//  annxud,    they    nuiy    be    considered    on    tlie    lines    of    the 
instalment  method  of  local  authorities.     (Chapter  XI.) 


A    SURPLUS    IN    THE    FUND  221 

7/  at  fie  nods  of  years,  a  sinking  fund  may  te  set  aside 
during  each  period  to  provide  tlie  proportion  of  tlie  loan 
repayable  at  the  end  of  each  period. 

If  at  2>cnods  of  years,  in  a  series,  a  sinking  fund  may  be 
provided  by  setting  aside  and  accumulating  equal  annual 
amounts  during  the  whole  period  in  order  to  provide  the 
amounts  repayable  at  the  end  of  each  period.  This  will 
apply  to  the  simultaneous  provision  out  of  profits  of  loans 
repayable  in  certain  priorities. 

Redemption  of  Loans  (Issued  as  Stock)  at  a  Premium  : 

If  the  premium  be  stated,  the  sinking  fund  instalment 
should  be  calculated  to  provide  that  amount  in  addition 
to  the  par  value,  and  there  is  not  any  change  in  the 
method  described. 

If  the  premium  depends  upon  the  price  at  the  date  of 
redemption,  and  cannot  be  accurately  estimated,  the 
annual  instalment  should  be  based  upon  the  par  value  of 
the  stock,  and  the  premium  provided  for,  as  and  when  it 
arises,  by  charging  it  to  revenue  account,  or  by  making 
prudent  provision  in  anticipation. 

Redemption  of  Loan  in  Part. 

The  trust  deed  may  provide  that  if  any  part  of  the  loan  be 
redeemed  out  of  the  fund,  the  interest  previously  paid  upon 
such  redeemed  loan  shall  be  added  to  the  fund,  although  the 
rate  of  interest  payable  to  the  loanholder  be  higher  than  the 
calculated  rate  of  accumulation  of  the  fund.  This  will  cause 
a  surplus  in  the  fund  over  the  calculated  amount,  which  will 
liave  the  eifect  of  anticipating  the  final  matvirity  of  the  fund, 
whether  the  loan  is  repayable  on  a  specified  date  or  by  the 
accumulation  of  a  stated  instalment.  The  possibility  of 
making  any  provision  for  such  an  event  when  calculating  the 
original  instalment  in  the  case  of  an  ordinary  sinking  fund 
will  depend  upon  the  circumstances  of  each  individual  case. 

Cessation  of  Annual  Contributions.  Instead  of  making 
the  adjustment  by  spreading  any  surplus,  however  arising, 
equally  over  the  unexpired  portion  of  the  repayment  period,  it 
may  be  provided  that  the  amount  in  the  fund  shall  continue 
to  accumulate,  and  the  original  instalments  be  annually  paid 
in,  until  such  time  as  the  fund  is  of  such  an  amount  that  the 


222    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

present  investments  and  the  accumulations  of  the  annual  income 
to  be  received  therefrom  in  future  will  be  sufficient,  without 
any  further  instalments,  to  provide  the  amount  of  loan  re- 
payable.    (See  Article  11  (2)  County  Stock  Reo-ulations,  1891.) 

CoxTixuATiox  OF  INSTALMENTS,  It  may  be  provided  that 
the  original  instalment  shall  continue  to  be  set  aside  and  added 
to  the  fund  until  the  loan  is  ultimately  repaid,  notwithstanding  : 

(1)  The  withdrawal  of  any  part  of  the  loan  from  the  operation 

of  the  sinking  fund  by  reason  of  its  being  converted  into 
ordinary  share  capital  or  stock. 

(2)  The  sale  of  any  part  of  the  assets  forming  part  of  the 

security  for  the  loan,  and  the  payment  of  the  proceeds 
into  the  fund. 

(3)  Any  other  cause  operating  to  produce  a  surplus  in  the 

fund  or  to  accelerate  the  date  of  maturity  of  the  fund. 

In  such  cases  it  may  be  necessary  to  determine  the  reduced 
period  of  redemption  which  may  be  ascertained  by  one  or  other 
of  tlie  methods  described. 


THE    RATE    OF    ACCUMULATION  223 


CHAPTER  XIX. 

SINKING  FUND  PIIOBLEMS,    RELATING   TO 
THE  BATE  PER  CENT., 

OF    INCOME    UPON    THE    PRESENT    INVESTMENTS    REPRESENTING 
THE   AMOUNT   IN    THE   FUND;    AND   ALSO   THE   FUTURE   RATE    OF 
ACCUMULATION   OF   THE   FUND. 

Variation  A,  in  which  there  is  a  variation  in  the 

RATE    OF    accumulation    WITHOUT    ANY    VARIATION    IN    THE 

rate  of  income  upon  the  present  investments,   or  in 
the  period  of  repayment.  statement  xix.  b. 

Summary  of  the  methods  of  adjustment.  General  con- 
siderations AS  to  variations  in  the  rate  per  cent,  to  be 
treated  in  detail  in  the  following  chapters.  The 
deductive  method.  Statement  showing  the  final  re- 
payment OF  THE  loan  by  THE  OPERATION  OF  THE  AMENDED 
ANNUAL  INSTALMENT. 


Summary  of  the  methods  of  adjustment. 

(7)  The  dedaicti've  method,  as  summarised  helotv,  is  of  wider 
application  than  the  variation  in  the  rate  of  accumulation  only, 
and  has  been  so  uwrded  that  it  may  he  treated  as  the  standard 
method  relating  to  all  variations.  Statement  XIX.  A. 

{II)  The  direct  method,  without  calculation,  as  summarised 
at  the  head  of  Chaper  XX,  will  not  apply  to  this  variation. 

{Ill)  The  annual  increment  {balance  of  loan)  method,  as 
summarised  at  the  head  of  Chapter  XXII,  may  be  used,  but 
ivill  not  be  applied  to  the  example  under  revieiv.  The  method 
of  finding  the  amended  annual  increment  is  shown  in  Calcula- 
tion {XIX)  5. 

{IV)  The  annual  increment  (ratio)  method,  as  summarised 
at  the  head  of  Chapter  XXIII.  Statement  XXII.  C. 

Note.  The  terms  used  in  the  following  summary  are  fully 
explained  at  the  head  of  Chapter  XXII.  In  all  the  above 
methods,  it  is  imperative  that  the  rate  of  aecumulation  and  of 
income  from  investments  be  uniform  during  the  whole  of  the 
une.vpired  or  substituted  portion  of  the  repayment  period. 


224    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Summary  of  the  deductive  method,  of  ascertaining  the 
amended  annual  sinhinc)  fund  instalment  due  to  a  variation  in 
the  rate  yer  cent,  of  accumulation,  accompanied  by,  or  without, 
any  variation  in  the  rate  of  income  to  he  received  upon  the 
present  investments  repi-esenting  the  fund,  and  also  due  to  any 
variation  in  the  period  of  repayment,  or  any  combination  of  the 
above  factors.  Statement  XIX.  A. 

(1)  Ascertain   the    value   of   the   present  investments   in   the 

manner  already  described,  and  also  the  amount  of  the 
present  annual  incowe  yielded  by  such  investments,  up 
to  the  time  of  malting  the  adjustment. 

(2)  To  the  present  annual  income,    so  ascertained,   add   the 

present  or  original  annual  instalmeyit  which  has  been  set 
aside  and  added  to  the  sinl^ing  fund  up  to  the  time  of 
making  the  adjustment . 

[3]  The  total  so  obtained  is  the  present  annual  increment  of 
the  fund. 

{4)  Ascertain,  or  estimate,  the  rate  per  cent,  at  which  the 
fund  will  accumulate  in  future  (the  future  rate). 

(5)  Calculate  {in  one  sum  or  separately)  the  amount  of  the 

present  annual  increment  found,  as  in  (3),  for  the  number 
of  years  in  the  ^inexpired  or  substittited  period  of  repay- 
ment, at  the  future  rate  of  accumulation  fxed  in  (4). 

Calculations  {XIX)  1  and  2. 

(6)  The  amount  or  amourits,  so  ascertained,  will  represent  the 

portion  of  original  loan  which  will  be  provided  at  the 
end  of  the  original  or  varied  period  of  repayment . 

(7)  To    this    amount   add    the   value    of    the    present   invest- 

ments, as  ascertained  in  (1),  and  deduct  the  sum  from  the 
amount  of  the  origimd  loan. 

(S)  The  remainder  represents  the  portion  of  original  loan 
which  is  now  ^inprovided  for  by  the  present  investments 
and  the  future  accumulation  of  the  present  annual  incre- 
inent  found  in  {3). 

(9)  Calculate  the  additional  annual  sinMng  fund  instalment 
wliich,  at  the  future  rate  of  accumulation^  estimated  as 
in  [4),  will  amount  to  the  balance  of  loan  found  in  (8) 
at  the  end  of  the  unexpired  or  substituted  period  of 
repayment.  Calculation  [XIX)  3. 


THE    RATE    OF    ACCUMULATION  225 

(10)  This  additional  annual  instahnent,  added  to  the  ^present 

annual  increment  found  in  {3)  gives  the  same  future  or 
amended  annual  increTnent,  which  is  found  by  direct 
calculation  hy  the  annual  increTnent  {ratio)  Tnethod. 

(11)  From  the  future,  or  amended  annual  increment,  so  ascer- 

tained, deduct  the  future  annual  income  from  the  present 
investments ;  and  the  remainder  is  the  future  or  amended 
annual  instalment  to  be  charged  to  revenue  or  rate  in 
substitution  for  the  present  or  original  annual  instalnnent. 

(12)  Prepare  a,  statement  showing  the  final  repayment  of  the 

loan  by  the  operation  of  the  fund  under  the  amended 
conditions .  Statement  XIX.  B. 

(13)  Prepare  a  pro  forma  account  sltowing  the  amount  which 

should  be  in  the  fund  at  the  end  of  each  year  of  the 
unexpired  or  substituted  period  of  repayment. 

Pro  forma  Account,  No.  7. 

Memo.  The  above  method  is  worded  to  apply  to  a  reduction 
in  the  rate  of  accumulation  or  other  factor,  hut  it  unll  apply 
equally  to  an  increase  in  suck  factors  with  very  little  modi f ca- 
tion. It  should  be  compared  with  the  deductive  method 
summarised  at  the  head  of  Chapter  XXIV . 

General  Considerations  as  to  the  Rate  per  cent. 

Having  described  the  various  methods  of  dealing  with 
problems  arising  out  of  a  deficiency  or  a  surplus  in  the  sinking 
fund,  further  questions  will  now  be  considered  in  connection 
with  the  rate  per  cent.,  beginning  Avith  cases  in  which  it  is 
anticipated  that  the  original  estimated  rate  of  accumulation 
will  not  be  realised  in  future.  This  is  mainly  due  to  a 
fluctuation  in  the  money  market  of  a  more  or  less  permanent 
character  affecting  the  future  return  on  all  investments. 
Questions  will  also  arise  in  consequence  of  a  reduction  in  the 
rate  of  income  to  be  received  in  future  on  investments  already 
made,  as  was  the  case  in  1888,  when,  under  Mr.  Goschen's 
Finance  Act,  the  rate  of  Consols  was  reduced  from  3  per  cent, 
to  2f  per  cent,  for  15  years,  after  which  a  further  reduction  to 
2i  per  cent,  took  place.  Other  causes  may  operate  in  a  similar 
manner,  especially  in  the  case  of  commercial  and  financial 
undertakings. 

The  problem  will  differ  according  as  the  variation  in  the 
original  conditions  affects:  — 


226         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 

(1)  The  rate  of  accumulation  anticipated  to  be  realised  on 

tlie  investment  of  future  accretions  to  the  fund. 

Variation  A. 

(2)  The  rate  of  income  to  be  received  on  the  present  invest- 

ments representing  the  fund.  Variation  B. 

(3)  Both  the  above  rates  in  combination.  Variation  C. 

In  making  the  adjustments  it  will  at  times  be  difficult  to 
forecast  accurately  the  future  rate  of  income  to  be  received  on 
the  present  investments.  In  such  cases  it  is  wise  to  form  a 
conservative  estimate  of  the  future  rate  and  fix  it  on  the  low 
side;  or  to  take  a  slightly  lower  rate  of  accumulation  and 
therebv  increase  the  annual  instalment  to  be  charged  to  revenue 
or  rate  account.  In  discussing  the  following  variations  it  will 
be  assumed  that  although  the  future  rate  of  income  to  be 
received  upon  the  present  investments  will  change,  yet  it  will 
be  uniform  during  the  whole  of  the  unexpired  repayment 
period.  But  cases  may  arise  in  which  this  will  not  be  so,  but 
in  which  the  rate  of  income  will  again  vary,  during  the  term, 
in  a  definite  manner  laid  down  in  advance,  as  in  the  case  of 
Consols  previously  referred  to.  A  variation  of  this  nature, 
occurring  during  the  unexpired  portion  of  the  repayment  period, 
will  be  deferred  to  Chapter  XXYII.  When  considering 
Variation  B  (rate  of  income  only)  in  Chapter  XX,  it  will  be 
found  that  the  future  rate  of  accumulation  is  the  most 
important  factor  in  the  adjustment,  although  it  may  not  be  the 
greater  as  regards  the  actual  amount  of  money  involved. 

The  following  discussion  will  be  confined  to  a  reduction 
only  in  both  the  above  rates  per  cent.,  but  it  should  be  borne 
in  mind  that  the  method  to  be  adopted  and  described  will  apply 
equally  to  an  increase  in  both  rates  or  to  an  increase  in  one  and 
a  decrease  in  the  other.  This  will  be  better  appreciated  after 
considering  the  methods  of  making  the  adjustment  by  the 
annual  increment  (ratio")  method. 

Any  deficiencv  in  the  fund  at  the  time  of  making  the 
enquiry,  and  arising  out  of  a  reduction  in  the  rate  of  income 
received  from  investments  previously  made,  or  from  other 
causes,  will  not  affect  the  present  method  of  colrulation.  Any 
such  deficiencv  may  or  may  not  be  discovered  on  ascertaining 
the  present  position  of  the  fund  as  described  in  the  previous 
chapter.  The  following  method  differs  from  the  one  there 
described,  in  that,  in  the  present  example,  the  basis  of  the 
adjustment  is  the  value  of  the  present  investments,  and  not  the 
omonnt  to  which  thev  Avill  accumulate  at  the  end  of  the  term. 


THE    RATE    OF    ACCUMULATION  227 

In  dealing-  with  a  deficiency,  it  was  assumed  that  there 
would  not  be  any  variation  in  the  rate  of  accumulation,  whereas 
in  the  present  example  the  reduction  in  the  rate  of  accumula- 
tion is  the  cause  of  the  rectification  under  discussion. 

In  an  actual  enquiry  of  this  nature,  the  amount  in  the  fund 
at  the  end  of  the  12tli  year,  as  shown  by  the  records,  would  most 
probably  be  compared  with  the  calculated  amount  which  should 
be  in  the  fund  according  to  the  pro  forma  account,  and  the 
deficiency  or  surplus  thereby  ascertained,  but  it  is  not  absolutely 
necessary  to  do  this.  The  important  factors  are,  the  value  of 
the  present  investments,  the  future  income  they  may  be 
expected  to  produce,  and  the  rate  of  accumulation  which  will  be 
yielded  by  the  investment  of  the  future  accretions  to  the  fund. 
In  this  connection  Chapter  XIY,  dealing  generally  with  the 
present  investments  and  the  annual  increment  should  be 
consulted,  especially  as  to  the  meaning  of  the  term  "  present 
investments."  The  deductive  method  will  apply  to  the  rectifi- 
cation of  a  present  deficiency  or  surplus  combined  with  a 
variation  in  the  future  rates  of  income  or  accumulation,  because 
in  this  case  the  enquiry  is  based  upon  the  value  of  the  invest- 
ments now  representing  the  fund ;  and  the  method  of  approach- 
ing the  problem  is  not  altered  because  that  value  is  greater  or 
less  than  the  amount  which  should  be  in  the  fund  according 
to  the  original  calculation,  and  as  shown  by  the  pro  forma 
account.  The  method  about  to  be  described  will  show  the 
amended  annual  instalment  to  be  charged  to  revenue  or  rate, 
based  upon  the  present  state  of  the  fund,  but  if  it  be  required 
to  allocate  this  as  between  a  present  deficiency  or  surplus  and 
the  future  reduction  in  the  rates  of  income  or  accumulation,  it 
will  be  necessary  to  make,  first,  the  calculation  as  to  the 
deficiency  or  surplus,  as  already  described,  followed  by  the 
enquiry  as  to  the  increased  annual  instalment  due  solely  to  the 
fall  in  the  rate  or  rates  per  cent. 

Details  of  the  Sinking  Fund.  The  sinking  fund  which 
will  be  used  to  illustrate  all  problems  relating  to  a  variation  in 
the  rate  per  cent,  will  apply  to  a  loan  of  £26,495,  repayable  at 
the  end  of  a  period  of  25  years,  requiring  an  annual  instalment 
of  £680234  to  be  set  aside  and  accumulated  at  3|  per  cent. 
[Calculation  (XV)  1],  and  it  will  in  all  cases  be  assumed,  as 
when  considering  the  rectification  of  a  surplus,  that  at  the  end 
of  the  12th  year  the  fund  stands  at  the  proper  calculated 
amount  of  £9932-74,  as  found  by  Calculation  (XY)  2.  This 
sum  is  represented  by  investments  worth  that  amount    which 


228    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

have  up  to  the  present  yielded  an  annual  income  at  the  rate  of 
3^  per  cent,  per  annum,  being  the  original  estimated  rate  of 
accumulation  upon  which  the  above  instalment  was  based. 
This  sum  of  £9932" 74,  if  accumulated  at  the  above  rate  of 
3|  per  cent.,  will  provide  for  the  repayment  of  £15534"38  of 
original  loan  at  the  end  of  25  years,  as  found  by  Calculation 
(XVII)  2. 

Variations  in  the  Rate  Per  Cext.,  to  be  considered  in 
Detail.  In  order  to  illustrate  the  problems  to  be  discussed  in 
this  and  following  chapters  three  variations  from  the  original 
conditions  as  regards  the  rate  of  accumulation  of  3^  per  cent,  will 
be  considered.  In  fixing  this  rate  percent,  in  the  first  instance  it 
was  assumed  that  it  would  continue  to  be  received  upon  the 
whole  of  the  accumulations  of  the  fund  during  the  whole  of  the 
repayment  period  of  25  years.  If  this  anticipation  had  been 
realised  the  rate  per  cent,  of  income  upon  investments  and  the 
rate  per  cent,  of  accumulation  would  have  been  the  same  in  all 
cases,  namely,  3^  per  cent.,  and  the  fund  would  have  pursued 
its  calculated  course  until  maturity. 

In  the  three  examples  about  to  be  considered  a  gradual 
decrease  in  the  rate  of  income  from  investments,  as  well  as  in 
the  rate  of  accumulation,  will  be  assumed  to  occur  between  each 
set  of  conditions;  but  when  comparing  the  several  results  in  a 
later  chapter  they  will  be  considered  only  as  regards  the  altera- 
tion in  the  rate  of  accumulation  as  follows  :  — 

Future  rate  of 

income  Future 

on  present  rate  of 

Chapter.       Variation.  Compared  witli  investments.  accumulation. 

XIX  A    Original  conditions     unaltered      reduced 

XX  B  Variation  A  reduced         unaltered 

XXI  C  Variation  A  reduced         reduced 

The  paramount  importance  of  the  rate  of  accumulation  in 
such  problems  has  already  been  referred  to,  and  it  will  be 
noticed  from  the  above  table  that  Variations  (A)  and  (C)  alone 
contain  any  variation  in  that  rate.  The  following  details  as 
to  each  variation  are  given  for  convenience  of  reference  and 
comparison  :  — 

Chapter  XIX.   Variation  {A)  in  the  rate  of  accunndation  onh/. 
Compared  with  the  conditions  at  the  time  the 

original  calculation  was  made. 
In  this   example   the   rate   of   accumulation    is 
reduced  from  3^  to  3  per  cent.,  but  the  rate 
of    income    upon    the   present    investments 
remains  at  3^  per  cent. 


THE    KATE    OF    ACCUMULATION  229 

Chapter  XX.  Variation  {B)  in  the  rate  of  income  upon  tlte 
present  investTnents  only. 
Compared  with  the  couditions  iu  Variation  (A). 
In  this  example  the  rate  of  accumulation  is 
unaltered,  and  remains  at  3  per  cent.,  but 
the  rate  of  income  upon  the  present  invest- 
ments is  redviced  from  3^  to  3  per  cent. 

Chapter  XXI.  Variation  (C)  in  the  rate  of  accumulation,  as 
well  as  in  the  rate  of  income  upon  the 
present  investments. 

Compared  with  the  conditions  in  Variation  (A) . 

In  this  example  the  rate  of  accumulation  is 
reduced  from  3  to  2|  per  cent.,  and  the  rate 
of  income  upon  the  present  investments  is 
reduced  from  3|  to  3  per  cent. 

These  variations  will  now  be  examined,  and  will  be  treated 
as  independent  problems  instead  of  variations  of  the  same  fund. 
This  procedure  involves  a  certain  amount  of  repetition,  but  is 
adopted  in  order  to  emphasize  the  principles  involved,  with  the 
view  of  finding  a  shorter  method  of  making  the  adjustments. 
There  is  also  a  further  advantage,  namely,  that  each  problem 
may  be  studied  separately  so  that  any  cases  occurring  in  actual 
practice  may  be  referred  to  a  similar  example  completely 
worked  out  in  detail. 

It  will  be  noticed  on  referring  to  the  above  details  and  to  the 
summary  of  results  given  in  Chapter  XXI,  Statement  XXI,  C, 
that  the  above  variations  are  not  isolated  cases  without  any 
connection.  They  are  intimately  related  by  design,  and  form  a 
series  commencing  with  the  original  conditions  and  leading  by 
successive  stages  to  Variation  C  (rate  of  income  and  accumula- 
tion). When  considering  the  derivation  of  a  rule  and  formula 
relating  to  the  adjustment  of  a  sinking  fund  in  consequence  of  a 
simultaneous  variation  in  the  rates  per  cent,  of  accumulation 
and  income  on  investments  these  variations  will  be  combined, 
and  in  one  instance  (Calculation  XXII,  C),  Variation  A  will  be 
inverted  to  serve  as  an  example  of  an  increase  in  the  rate  per 
cent,  of  accumulation. 

Any  decrease  in  the  rate  of  income  yielded  by  the  present 
investments  or  by  the  future  investments  of  the  annual 
accretions  to  the  fund  will  have  the  effect  of  reducing  the  sum 
to  Avhich  the  fund  will  amount  at  the  end  of  the  repayment 
period.  The  amount  of  such  deficiency  will  depend  upon  the 
actual  rates  to  be  received   in   future   as   compared   with   the 


230    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

original  rate  of  accumulation,  namely,  13^  j)er  cent.  It  is 
necessary  so  to  adjust  the  sinking  fund  that  the  deficiency  due 
to  a  fall  in  the  rate  either  of  income  or  of  accumulation  shall 
not  only  be  made  good,  hut  be  spread  equally  over  the  remain- 
ing 13  years,  by  increasing  the  original  sinking  fund  instalment 
by  such  an  annual  amount  as  will  be  sufficient  for  the  purpose. 

The  Deductive  Method.  In  order  to  ascertain  the  amount 
by  which  the  annual  instalment  should  be  increased,  the  present 
sinking  fund  factors  may  be  reduced,  either  to  terms  of  present 
value  or  to  equivalent  amounts  of  original  loan  repayable  at 
the  end  of  the  25  years,  but,  as  in  the  former  example,  it  is 
preferable  to  deal  with  the  figures  representing  equivalent 
amounts  of  loan. 

In  each  of  the  above  variations  the  common  factors  are  :  — 

(1)  A  sum  of  £9932' 74  standing  to  the  credit  of  the  fund  at 

the  end  of  the  12th  year,  which  is  invested  and  expected 
to  realise  that  sum  at  the  end  of  the  repayment  period. 

(2)  The  income  arising  from  the  above  present  investments. 

(3)  The  original  annual  instalment  of  £680234  to  be  set  aside 

for  the  unexpired  term  of  13  years,  and  which  will  also  be 
invested  each  year. 

(4)  The  income  to  be  received  annually  from  (2)  and  (3)  when 

invested. 
Items  (2)  and  (3)  constitute  the  present  annual  increment  of  the 
fund,  as  described  in  Chapter  XIV  and  in  Chapter  XXII. 

In  each  case  the  original  annual  instalment  of  £680'234  will 
be  supplemented  by  an  additional  annual  instalment  to  be  ascer- 
tained, and  which,  added  to  the  present  annual  increment,  Avill 
give  the  future  or  amended  annual  increment  of  the  fund.  The 
method  of  approaching  the  solution  of  the  problem  is  the  same 
in  each  variation. 

A  statement  will  be  prepared  similar  to  XIX,  A,  showing 
the  position  of  the  fund  at  the  end  of  the  12th  year,  when  the 
assumed  necessity  arises  to  make  the  adjustment  due  to  a 
change  in  the  rate  per  cent,  either  of  income  or  accumulation 
or  both.  This  statement  will  commence  with  the  amount  now 
in  the  fund,  which  will  be  included  at  its  present  value  without 
accumulation.  This  is  equivalent  to  deducting  that  amount 
from  the  original  loan,  leaving  the  balance  to  be  provided  by 
the  accumulation  of  the  future  or  amended  annual  increment 
which  is  composed  of  the  future  income  from  the  present 
investments  and  the  amended  annual  instalment. 


THE    RATE    OF    ACCUMULATION  231 

This  is  a  departure  from  the  procedure  followed  previously 
in  dealing  with  a  surplus  or  a  deficiency  in  the  fund,  in  which 
cases  there  was  not  any  change  in  either  of  the  rates  per  cent. 

The  above  Statement  XIX,  A,  will  next  include  the  present 
annual  increment  consisting  of  the  income  from  the  present 
investments  prior  to  the  variation  occurring,  and  also  the 
original  annual  instalment.  Both  these  annual  sums  will  be 
converted,  by  calculation  at  the  future  accumulation  rate,  into 
equivalent  amounts  of  original  loan  repayable  at  the  end  of  the 
unexpired  period.  The  balance  will  represent  the  amount  of 
original  loan  for  which  further  provision  has  to  be  made  caused 
by  the  decrease  in  the  rates  of  income  or  of  accumulation,  and 
from  this  balance  of  loan  the  required  additional  annual 
instalment  may  be  ascertained  on  standard  calculation  form. 
No.  3x.  There  is  a  diierence  in  the  method  of  treating  the 
income  from  investments  in  Statements  XX,  A,  and  XXI,  A, 
as  compared  with  Statement  XIX,  A,  but  they  may  all  be 
treated  by  the  deductive  method  summarised  at  the  head  of  this 
chapter.  A  further  statement  similar  to  XIX,  B,  is  then 
prepared  in  each  case  showing  how  the  fund  will  ultimately 
work  out  to  repay  the  full  amount  of  the  loan  at  the  end  of  the 
original  repayment  period. 

Having  ascertained  the  future  or  amended  annual  instalment 
in  each  case  by  the  deductive  method,  the  results  will 
afterwards  be  used  to  derive  therefrom  a  simple  rule  and 
formula  by  which  to  make  the  calculation  by  direct  reference 
to  the  published  tables  or  formulae.  It  will  then  be  found  that 
by  taking  the  present  annual  increment  as  the  prime  factor 
instead  of  the  annual  instalment,  all  such  variations  may  be 
divided  into  two  classes  depending  entirely  upon  the  rate  of 
accumulation.  In  variations  similar  to  A  and  C,  in  which  the 
rate  of  accumulation  is  reduced  or  increased,  a  calculation  must 
be  made  by  means  of  the  tables  or  formula,  but  in  variations 
similar  to  B,  where  there  is  a  variation  in  the  rate  of  income 
only,  the  rate  of  accumulation  remaining  unaltered,  the 
amended  annual  instalment  may  be  ascertained  without 
calculation.  This  method  is  shown  in  Statement  XX.  C.  called 
"  the  direct  method,"  and  it  may  appear  superfluous  to  include 
the  longer  deductive  method  shown  in  Statement  XX,  A. 

It  is  necessary,  however,  to  state  that  in  all  cases  the  income 
from  investments  has  been  treated  as  being  received  annually, 
whereas  in  all  probability  it  would  be  received  half-yearly. 
The  difference  between  an  annual  and  a  semi-annual  accumula- 
tion has  been  pointed  out  at  the  end  of  Chapter  V,  giving  also 


232    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

the  methods  to  be  adopted  in  the  case  of  any  periodic  accumula- 
tion other  than  annual. 

The  Annual  Increment  (Ratio)  Method.  It  has  already- 
been  stated  that  instead  of  using  the  above  deductive  method 
the  same  result  may  be  obtained  by  direct  calculation  by  means 
of  a  formula  and  rule.  This  will  be  fully  described  in 
Chapter  XXII  (Calculation  XXII,  C),  which  shows  the  future 
or  amended  annual  increment  of  £1060474  as  found  by  the 
above  deductive  method.  From  the  amended  annual  increment 
so  found  the  future  or  amended  annual  instalment  niay  be 
obtained  by  deducting  therefrom  the  future  annual  income 
from  the  present  investments.  This  ratio  method  by  direct 
calculation  will  also  apply  to  Variation  C,  where  there  is  also  a 
change  in  the  rate  of  accumulation,  but  in  the  case  of 
Variation  B,  in  which  the  rate  of  accumulation  remains  un- 
altered, the  amended  annual  instalment  can  be  ascertained  by  a 
much  more  direct  method  without  calculation,  as  explained 
above. 

The  method  of  finding  the  amended  annual  instalment  is 
shown  in  the  following  Statement  XIX,  A,  and  the  final 
repayment  of  the  loan  thereby  is  shown  in  Statement  XIX,  B, 
and  in  the  pro  forma  account  'No.  7. 


The  Rate  per  cent.  Statement  XIX.  A. 

The  Deductive  Method. 

Variation  A,  rate  of  accumulation  only. 

Showing  the  method  of  adjusting  the  annual  instalment^  in 
consequence  of  a  variation  in  the  rate  of  accumulation 
without  any  variation  in  the  rate  of  income  from  the 
present  investments,  or  in  the  period  of  repayment. 

This  example   is   compared   with   the   conditions   at   the 
time  the  original  calculation  was  made. 

Conditions  before  adjustment  (at  end  of  12th  year) 

Amount  of  loan  repayable  in  25  years £26,495 

Amount  in  the  fund  (at  end  of  12th  year) £9932-74 

Present  annual  income  (previously)  received  there- 
from, at  3^  per  cent,  per  annum       £347*648 

Present  annual  instalment,  to  be  accumulated  for 

13  years  at  3|  per  cent £6(S0"234 

Present  annual  increment £1027'882 


THE    RATE    OF    ACCUMULATION  233 

Variation  from  the  above  conditions  : — 

The  rate  of  accumulation  of  the  fund  is  reduced  from  3|  to 
3  per  cent. 

Equivalent 

amount  of 

original  loan. 

Present  investments    (at    end     of     12th     year), 

representing  the  amount  now  in  the  fund       ...       £9932"74 

Present  annual  income  from  investments  ;— 

Amount  of  an  annuity  of       £347648 


accumuhited  for  13  years,  at  3  per  cent. 

Calculation  (XIX)  1       £5429-49 


Original  annual  instalment : — 

Amount  of  an  annuity  of £680"234 


accumulated  for  13  years,  at  3  per  cent. 

Calculation  (XIX)  2     £10623-75 


Present  annual  increment    £1027882 


Provision  already  made  will  repay  loan  of     £25985-98 

Additional  annual  instalment  :— 

Balance,  heing  amount  of  original  loan  un- 
provided for  owing  to  the  above  decrease  in 
the  rate  of  accumulation  requiring  an 
additional  annual  instalment,  to  he  set  aside 
and  accumulated  for  13  years  at  3  per  cent.  £509-02 
Calculation  (XIX)  3     £32-592 

Amount  of  original  loan £26495-00 


Amended  annual  increment,   being  :  — 

Income  from   investments        ...  £347-648 
Amended  annual  instalment  ...       712-826 


£1060-474 


234 


REPAYMENT  OF  LOCAL  AND  OTHER.  LOANS 


The  Rate  per  cent. 


Statement  XIX.  B. 


Variation  A,  rate  of  accumulation  only. 

Showing  the  final  repayment  of  the  loan,  by  tlie  operation  of 
the  sinking  fund  after  making  the  adjustment  in  the 
annual  instalment,  consequent  upon  a  variation  in  the  rate 
of  accumulation,  without  any  variation  in  the  rate  of 
income  upon  the  present  investments,  or  in  the  period  of 
repayment. 


Present  investments  (at  end  of  12th  vear) 


Equivalent 

amount  r{ 

original  loan. 

£9932-74 


Amended  annual  increment  :— 

Original  annual  instalment  ... 
Additional  annual  instalment 


£680-234 
32-592 


Total  out  of  revenue     £712'826 
Income    from    investments    347-648 


£1060-474 


Amount  thereof,  accumulated  for  13  years  at 

3  per  cent.  Calculation  (XIX)  4     £16562-26 


Amount  of  original  loan 


£26495-00 


THE    RATE    OF    ACCUMULATION 


235 


Pro  forma  Sinking  Fund  Account,  No.  7. 

A  Variation  in  tlie  Hate  of  Accumulatiou  only. 

Loan  of  £26,495  re'payahle  at  the  end  of  25  years. 

Showing  the  final  repayment  of  the  loan,  by  the  operation  of 
the  increased  annual  instalment  of  £712'826. 


Statement  XIX.  B. 


Rate  of  accumulation,  3  per  cent. 


Year. 

Amount  in 
the  fund 

at  beginning 
of  year. 

Income 
received  from 
investments 
31  per  cent. 

Annual        i 
sinking  fund 
instalment. 

received  from           Amount  in 
nvestments  made       the  fund 
after  l-2th  year             at  end 
3  per  cent.                of  year. 

Year. 

1 

1 

2 

2 

3 

3 

4 

The 

amount  in  the  fun( 

i  at  the  end  of 

4 

5 

the  12th  year, 

£9932-744,  is  the  correct 

5 

6 

calculated  amo 

unt,  as  sill 

own  by  Ca 

Icula- 

6 

7 

tion 

(XV)  2, 

and    by 

the-   pro 

forma 

8 

account,  No.  1 

,  Chapter 

XV. 

8 

9 

9 

10 

10 

11 

11 

12 

9932-744 

12 

13 

9932-744 

347-648 

712-826 

— 

10993-218 

13 

14 

10993-218 

347-648 

712-826 

31-814 

12085-506 

14 

15 

12085-506 

347-648 

712-826 

64-583 

13210-563 

15 

16 

13210-563 

347-648 

712-826 

98-335 

14369-372 

16 

17 

14369-372 

347-648 

712-826 

133-099 

15562-945 

17 

18 

15562-945 

347-648 

712-826 

168-906 

16792-325 

18 

19 

16792-325 

347-648 

712-826 

205-787 

18058-586 

19 

20 

18058-586 

347-648 

712-826 

243-775 

19362-835 

20 

21 

19362-835 

347-648 

712-826 

282-903 

20706-212 

21 

22 

20706-212 

347-648 

712-826 

323-204 

22089-890 

22 

23 

22089-890 

347-648 

712-826 

364-714 

23515-078 

23 

24 

23515-078 

347-648 

712-826 

407-470 

24983-022 

24 

25 

24983-022 

347-648 

712-826 

451-504 

26495000 

25 

236         REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 


CHAPTER  XX. 

SINKING  FUND  PROBLEMS,  RELATING  TO  THE 
RATES  PER  CENT.  OF  INCOME  AND  ACCUMULA- 
TION {Co7itinued) . 

Variation  13,  in  which  there  is  a  variation  in  the  rate 
OF  income  upon  the  present  investments  without  any 

VARIATION    IN    THE    RATE    OF    ACCUMULATION    OR    PERIOD    OF 

Repayment.  Statement  XX.  A. 

Summary  of  the  methods  of  adjustment.  The  deductive 
METHOD.  The  direct  method  without  calculation. 
The  annual  increment  (balance  of  loan)  method. 
Statement  showing  the  final  repayment  of  the  loan  by 

THE  operation   OF   THE  AMENDED   ANNUAL   INSTALMENT. 


Summary  of  the  methods  of  adjustment. 

(/)  The  deductive  tnethod,  as  summarised  at  the  head  of 
Chapter  XIX,  ivill  not  ayfly,  since  there  is  not  any  variation 
in  the  rate  of  accumulation.  The  following  adjustment  by  the 
deductive  method  is  only  of  academic  interest  and  has  hardly 
any  practical  value.  Statement  XX.  A. 

(IT)  The  direct  method,  without  calculation^  as  summarised 
below,   should  always  he  used  in  acttial  practice. 

Statement  XX,  C. 

{Ill)  The  annual  increment  [balance  of  loan)  method,  as 
summarised  at  the  head  of  Chapter  XXll,  may  be  used. 

Statement  XX.   D. 

{IV)  The  annual  increment  [ratio)  method,  as  summarised 
at  the  head  of  Chapters  XXIII,  XXV ^  and  XXVI,  will  not 
apply  to  tit  is  variation,  as  there  is  not  any  change  in  the  rate 
of  accumulation. 

Note.  The  terms  used  in  the  following  summary  are  fully 
explained  at  the  head  of  Chapter  XXII.  If  it  he  l-nown  or 
anticipated  that  the  rate  of  income  to  be  yielded  in  future  by 
the  present  investments  representing  the  fund,  toill  not  be 
uniform  during  the  whole  of  the  unexpired  portion  of  the 
repayment  period  the  above  methods  will  not  apply,   and  the 


THE    RATE    PER    CENT.    OF    INCOME  237 

adjustment  miist   he  viade   by   the    metlind  fully  described  in 
Chapter  XXV 17. 

vSuMMARY  OF  THE  DIRECT  METHOD  (tvithout  Calculation), 
of  ascertaining  the  amended  annual  sinTcing  fund  instalment 
due  to  a  variation  in  the  rate  of  income  yielded  by  the  'present 
investments  without  any  variation  in  the  rate  of  accumulation 
or  in  the  period  of  repayment.  StatCTnent  XX.  C. 

(i)  Having  ascertained  the  value  of  the  present  investTnents 
in  the  manner  already  described, 

(2)  Calculate  the  annual  income  previously  received  there- 

from during  the  expired  portion  of  the  original  repay- 
ment period  {the  present  annual  incoTne). 

(3)  Calculate    the    annual    income    expected    to    be    received 

therefrom  during  the  unexpired  portion  of  the  original 
repayment  period  at  the  future  rate  per  cent,  of  income 
{the  future  annual  income). 

(4)  Ascertain  the  decrease  or  increase  in  such  future  annual 

income  as  compared  ivith  the  annual  income  previously 

received. 
{6)  Add  to,   or  deduct  from,  the  original  annual  instalment 

the  anmial  decrease  or  increase  of  income  so  ascertained. 
{6)   The  result  is  the  amended  annual  instalment  to   be  set 

aside  out  of  revenue  or  rate  during  the  unexpired  portion 

of  the  original  repayment  period. 

(7)  Prepare  a  statement  showing  the  final  repayment  of  the 

loan  by  the  operation  of  the  sinking  fund  under  the 
amended  conditions.  Statement  XX.   B. 

(8)  Prepare  a  pro  forma  account  showing  the  amount  which 

should  be  in  the  fund  at  the  end  of  each  year  of  the 
unexpired  repayment  period. 

Pro  forma  account,  No.  8. 
The  amounts  in  the  fund  at  the  end  of  each  year  u'ill  be 
the  same  as  in  the  original  pro  forma  account  since  there 
is  not  any  variation  in  the  rate  of  accumulation  or  period 
of  repayment,  but  the  annual  increment,  although 
unaltered  will  have  a  different  origin.  Pro  forma 
account  A^o.  1,  Chapter  XV ^  will  not  apply  in  this  case, 
the  rate  of  acciimulation  being  3\  per  cent. 

The  Deductive  Method.  After  discussing  the  deductive 
method  of  finding  the  amended  annual  instalment  due  to  a 
change  in  the  rate  per  cent,  of  accumulation  (Variation  A)  in 


238    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Chapter  XIX,  a  summary  of  tlie  successive  stages  of  tlie  adjust- 
ment has  been  prepared  and  placed  at  the  head  of  that  chapter, 
and  it  is  therefore  only  necessary  to  refer  to  that  summary. 
Attention  has  already  been  drawn  to  the  general  considerations 
to  be  borne  in  mind  in  the  rectification  of  a  sinking  fund  in 
consequence  of  any  variation  in  the  rates  per  cent,  of  income  or 
accumulation.  The  variation  about  to  be  considered  is  based 
upon  the  same  imaginary  sinking  fund  as  Yariation  A  (rate  of 
accumulation),  details  of  which  are  given  in  the  previous 
chapter.  At  the  end  of  the  12th  year  the  sinking  fund  stands 
at  the  proper  calculated  amount  of  i69932'74,  as  found  by 
Calculation  (XV)  2.  But  whereas  the  conditions  in  Variation  A 
(rate  of  accumulation)  were  compared  with  the  original  condi- 
tions, the  present  Variation  B  (rate  of  income),  will  be  compared 
with  the  conditions  in  Variation  A  (rate  of  accumulation). 

The  rate  of  income  is  reduced  from  3^  to  3  per  cent.,  but 
the  rate  of  accumulation  is  unaltered,  and  remains  at  3  per  cent. 

It  has  been  stated  in  the  previous  chapter  that  the  future 
rate  of  accumulation  is  the  most  important  factor  in  the 
adjustment.  That  conclusion  was  based,  in  advance,  upon  the 
results  of  the  discussion  of  the  present  variation,  because, 
although  the  same  deductive  method  will  be  used  which  has 
been  applied  to  Variation  A  (rate  of  accumulation),  this  method 
is  quite  unnecessary  in  practice,  although  it  is  instructive  as 
illustrating  the  predominant  effect  of  the  variation  in  the  rate 
of  accumulation. 

It  will  be  found  that  when  the  variation  in  the  rate  per  cent, 
applies  only  to  the  rate  of  income  from  the  present  investments 
there  is  not  any  necessity  to  make  .any  calculation  whatever 
beyond  adding  to  the  original  annual  sinking  fund  instalment 
an  amount  equal  to  the  annual  loss  of  income  caused  by  the 
reduced  yield  per  cent,  of  the  present  investments,  or  by 
deducting  therefrom  any  increase  in  such  annual  income.  The 
remarks  in  tbe  previous  chapter,  as  to  the  three  variations  being 
derived  by  successive  stages  from  the  original  conditions  should 
be  carefully  remembered,  and  will  be  further  emphasised. 

The  original  and  varied  conditions  are  given  in  the  following 
Statement,  XX.  A.,  and  attention  is  again  drawn  to  the  fact 
that  in  this  case  also  the  income  from  investments  is  treated  as 
being  received  annually,  instead  of  semi-annually.  Two  state- 
ments will  be  prepared  exactly  similar  in  principle  to  those  in 
the  previous  chapter,  dealing  Avith  Variation  A  (rate  of 
accumulation),  showing  in  XX.  A.  the  deductive  method  of 
ascertaining  the  amended  annual  instalment,  and  in  XX.  B. 


THE    RATE    PER    CENT.    OF    INCOME  239 

the  final  repayment  of  the  loan  by  the  operation  of  the  sinking 
fund,  under  the  altered  conditions.  For  the  purpose  of  the 
comparison  to  be  made  later,  this  variation  will  also  be 
compared  with  the  original  conditions.  (See  Statements  XX.  A. 
and  XX.  B,  at  end  of  chapter.) 

The  Direct  Method  (without  calculation).  It  has  been 
pointed  out  in  the  previous  chapter  dealing  with  a  variation 
in  the  rate  of  accumulation  only  that  instead  of  making  use 
of  the  deductive  method,  there  described,  for  the  purpose  of 
ascertainino-  the  amended  annual  instalment,  the  same  resiilt 
may  be  obtained  by  direct  calculation  by  means  of  a  rule  and 
formula,  which  will  be  fully  described  in  Chapter  XXIII, 
namely,  the  annual  increment  (ratio)  method.  This  remark 
applied  to  Variation  A  as  compared  with  the  original  conditions 
in  which  there  is  a  reduction  in  the  rate  of  accumulation,  but 
without  any  variation  in  the  rate  of  income  from  investments. 
In  the  present  case,  Variation  B,  as  compared  with  the  condi- 
tions in  Variation  A  (rate  of  accumulation)  there  is  a  reduction 
in  the  rate  of  income  upon  the  present  investments,  without 
any  variation  in  the  rate  of  accumulation,  and  the  deductive 
method  will  again  be  used.  On  comparing  the  two  results,  it 
is  found  that  in  both  cases  the  future  or  amended  annual 
increment  is  £1060-474,  although  the  amended  annual  instal- 
ment is  increased,  namely,  from  £712'826  in  Variation  A  to 
£762"490  in  Variation  B.  The  difference  between  the  two 
amended  annual  instalments  is  £49' 664,  which  is  the  amount  by 
which  the  future  annual  income  in  Variation  A  is  reduced 
owing  to  the  fall  of  |  per  cent,  in  the  rate  of  income  to  be 
yielded  by  the  present  investments  nnder  the  altered  conditions 
of  Variation  B,  namelv,  from  £84T'648  in  Variation  A  to 
£297-984  in  Variation  B. 

This  proves  that  when  the  rate  of  accumulation  remains 
unaltered,  there  is  not  any  alteration  in  the  annual  increment, 
and,  further,  that  the  amended  annual  instalment  may  be 
ascertained  without  any  calculation  whatever,  by  merelv 
adding  to  tlie  present  annual  instalment  the  amount  of  the 
decrease  in  the  annual  income  to  be  received  from  the  present 
investments  under  the  altered  conditions,  and  the  same  applies 
equally  to  an  increase  in  the  rate  per  cent,  yielded  by  the 
present   investments. 

The  following  Statement  XX.  C.  illustrates  the  ndjustment 
by  the  direct  method,  without  calculation. 

Althoufrh  the  direct  method  of  finding  the  amended  annual 


240    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

instalment  will  be  sufficient  in  all  cases  where  there  is  not  any 
variation  in  the  rate  of  accumulation,  it  should  be  proved  by 
preparing  a  statement  similar  to  No.  XX.  B.  showing  the 
position  of  the  fund  and  the  final  repayment  of  the  loan  after 
making  the  adjustment.  The  rule  and  formula  to  be  described 
later  in  Chapter  XXIII  (the  annual  increment  (ratio)  method), 
by  which  the  future  or  amended  annual  increment  under  the 
altered  conditions  may  be  found  by  direct  calculation  from  the 
present  annual  increment  under  the  previous  conditions,  cannot 
obviously  be  applied  to  cases  in  which  there  is  not  any  variation 
in  the  annual  increment,  which  depends  entirely  upon  the 
rate  of  accumulation. 

In  the  previous  chapter  the  deductive  method  is  employed 
to  ascertain  the  amended  annual  instalment,  consequent  upon 
a  variation  in  the  rate  of  accumulation  only.  In  the  following 
Chapter  (XXI),  in  discussing  Variation  C,  it  will  be  seen 
that  this  deductive  method  is  also  available  for  ascertaining  the 
amended  annual  instalment  consequent  upon  a  variation  in 
the  rate  of  accumulation,  accompanied  by  a  variation  in  the  rate 
of  income  from  the  present  investments.  But  in  the  case  of  a 
variation  in  the  rate  of  income  only,  the  deductive  method  may 
be  replaced  by  one  much  simpler.  At  the  head  of  this  chapter, 
therefore,  although  reference  is  made  to  the  deductive  method 
as  summarised  in  Chapter  XIX,  the  direct  method  without 
calculation  has  been  treated  as  the  standard  method  to  be 
adopted  in  practice,  and  has  been  stated  in  summary  form. 

In  Chapter  XIX,  the  conditions  in  Variation  A  (rate  of 
accumulation)  are  compared  with  the  original  conditions,  and 
it  has  been  found  that  an  additional  annual  instalment  of 
£'32'592  is  required  to  compensate  for  the  decrease  in  the  rate 
of  accumulation.  Proceeding  to  Variation  B,  it  has  been  found 
that  although  the  rate  of  accumulation  remains  unaltered,  the 
rate  of  income  from  investments  is  reduced.  This  reduction  in 
income  requires  a  further  increase  in  the  annual  instalment  of 
£49'664.  It  is  now  possible  to  compare  the  amended  annual 
instalment  in  Variation  B,  with  the  annual  instalment  under 
the  original  conditions  as  follows:  — 

The  original  annual  instalment  was £680"234 

Additionnl    instalment   due  to  the  reduction  in  the 

rate  of  accunuilation.  Variation  A.  32*592 

Additional   instalment  due  to  the  reduction  in  the 

rate  of  income  from  investments.       Variation   B  49'664 


Amended  annual  instalment.  Variation  B     ,£762'490 


THE    RATE    PER    CENT.    OF    INCOME  241 

or  au  increase  of  £82*256,  but  on  comparing  tlie  annual 
increment  in  A'ariation  B  (rate  of  income),  with  tlie  annual 
increment  under  the  original  conditions,  it  is  increased  by  only 
£32-592,  namely,  from  £102T-8cS2  to  £1060-474.  This  further 
proves  that  so  long  as  the  rate  of  accumulation  remains 
unaltered  the  annual  increment  does  not  require  to  be  amended, 
but  if  the  portion  of  the  annual  increment  derived  from  outside 
investments  is  reduced,  oM-ing  to  a  fall  in  the  rate  of  income 
yielded  by  the  present  investments,  the  burden  must  be  borne 
by  the  other  partner,  namely,  the  revenue  or  rate  account  which 
provides  the  annual  instalment. 

Statement  XX.  D.  shows  the  method  of  making  the  adjust- 
ment by  the  annual  increment  (balance  of  loan)  method,  which 
will  be  fully  described  and  summarised  in  Chapter  XXII. 

The  Rate  per  cent.  Statement  XX.  A. 

The  Deductive  Method. 

Variation  B,  rate  of  income  only. 

vShowing  the  method  of  adjusting  the  annual  instalment  in 
consequence  of  a  variation  in  the  rate  of  income  upon  the 
present  investments  without  any  variation  in  the  rate  of 
accumulation  or  in  the  period  of  repayment. 

This  example  is  compared  with  the  original  conditions 
as   modified   by   Variation    A. 

Conditions  before  adjustment    (at  end  of  12th  year), 

Amount  of  loan  repayable  in  25  years £26,495 

Amount  in  the  fund  (at  end  of  12th  year) £9932-74 

Present  annual  income  (previously)  received  there- 
from, at  3|  per  cent,  per  annum     £347'648 

Present  annual  instalment,  to  be  accumulated  for 

13  years  at  3  per  cent £71-^'8^6 

Present  annual  increment £10604(4 

Variation  from  the  above  conditions  : — 

The  rate  of  income  yielded  by  the  present  investments  is 
reduced  from  3^  to  3  per  cent. 

Future  annual  income £297'984 

Eeduction  in  annual  income        49*664 

Increased  annual   instalment        49-664 

Future  annual  increment     1060-474 


242         REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 

Equivalent 

amount  of 

original  loan. 

Present  investments    (at     end     of     12th     year), 

representing  the  amovmt  now  in  the  fund     ...       £9932'74 

Future  annual  income  from  present  investments : — 

Amount  of  an  annuity  of       £297 '984 


accumulated  for  13  years,  at  3  per  cent. 

Calculation  (XX)  1       £4653-85 

Original  annual  instalment : — 

Amount  of  an  annuity  of £680"234 


accumulated  for  13  years,  at  3  per  cent. 

Calculation  (XIX)  2       £10623-75 

Additional  annual  instalment  (Variation  A):  — 
Amount  of  an  annuity  of       £32592 


accumulated  for  13  years,   at  3  per  cent. 

Calculation  (XIX)  3         £509-02 


Provision  already  made  will  repay  loan  of    £25719-36 

Additional  annual  instalment  required  : — 

Balance,  being  amount  of  original  loan  un- 
provided for  owing  to  the  above  decrease  in 
the  rate  of  income  from  investments  requir- 
ing an  additional  annual  instalment,  to  be 
set  aside  and  accumulated  for  13  years  at 
3  per  cent £775-64 

Additional  annual  instalment 

Calculation  (XX)  2  £49-664 


Amount  of  original  loan    £26495"00 


Amended  annual  increment,  hriiig  :  — 

Income  from  investments         £297-984 

Amended  annual  instalment 762-490 

£1060-474 


THE    RATE    PER    CENT.    OF    INCOME 


243 


The  Rate  per  cent. 


Statement  XX.  B. 


Variation  B,  rate  of  income  only. 

Showing  the  final  eepayment  of  the  loan,  by  the  operation  of 
the  sinking  fund  after  making  the  adjustment  in  the 
annual  instalment,  consequent  upon  a  variation  in  the  rate 
of  income  upon  the  present  investments  without  any 
variation  in  the  rate  of  accumulation,  or  in  the  period  of 
repayment. 


Present  investments   (at  end  of  12th  year) 


Equivalent 

amount  of 

original  loan. 

£9932-74 


Amended  annual  increment :  - 

Original  annual  instalment     £680'234 

Additional.  Variation  A        32-592 

ditto.  Yariation  B        49-664 


Total  out  of  revenue 
Income  from  investments 


£762-490 
297-984 


£1060-474 


Amount  thereof,  accumulated  for  13  years  at 

3  per  cent.  Calculation  (XX)  3     £1656226 


Amoiint  of  original  loan 


£26495-00 


Amended  annual  instalment 


£762-490 


244    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


The  Rate  per  cent.  Statement  XX.  C. 

The  Direct  Method  (without  calculation). 

Variation  B,  rate  of  income  only. 

Showing  the  method  of  adjusting  the  annual  instalment  in 
consequence  of  a  variation  in  the  rate  of  income  upon  the 
present  investments  without  any  variation  in  the  rate  of 
accumulation  or  in  the  period  of  repayment. 

E-equired  the  amended  annual  instalment,  to  be  set  aside  and 
accumulated  as  a  sinking  fund  to  compensate  for  a 
reduction,  from  3^  to  3  per  cent.,  in  the  rate  of  income 
to  be  received  from  the  present  investments,  valued  at 
£9932' 74.     Rate  of  accumulation  3  per  cent. 

Annual  sinking  fund  instalment,  at  date  of  adjust- 
ment as  calculated  or  as  ascertained  in 
Variation  A.  Statement    XIX.  B.        £712-826 

Add  decrease  in  annual  income  from  investments 

at  3i  per  cent £347-648 

at  3  per  cent 297984 

£49-664 


Amended  annual  instalment       £762-490 


Memo.  In  the  case  of  an  increase  in  the  amount  of  the 
future  annua]  income,  such  increased  income  should  he  deducted 
from   the   oii<j-in:il    anmin]    instalment. 


THE    RATE    PER    CENT.    OF    INCOME  245 


The  Rate  per  cent.  Statement  XX.  D, 

The  Annual  Increment  (balance  of  loan)  Method. 

Variation  B,  rate  of  income  only. 

To  find  tlie  amended  annual  sinking  fund  instalment  consequent 
upon  a  variation  in  the  rate  of  income  upon  the  present 
investments,  without  any  variation  in  the  rate  of 
accumulation,  or  in  the  period  of  repayment. 

Eate  of  income  from  investments  reduced  from  ^  to  3  per  cent. 

Eate  of  accumulation,  3  per  cent. 

For  Eule,  see  Chapter  XXII. 

Amount  of  original  loan  (25  years) £2649500 

deduct  amount  in  the  fund  at  the  end  of  the 

12th  year £9932-74 


Balance  of  loan     £1656226 


Amended  annual  increment,  to  be  added  to  the 
fund,  and  accumulated  at  3  per  cent.,  to 
provide  this  amount  at  the  end  of  13  years 

Calculation   (XX)   4     £1060-474 

deduct  income  to  he  received  from  the  present 

investments  (£9932-74)  at  3  per  cent.       £297-984 

Amended   annual    instalment £(62-490 


246    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Pro  forma  Sinking  Fund  Account,  No.  8. 

A  Variation  in  the  rate  of  Income  upon  tlie  present  Investments. 

Loan  of  £26,495,  repayable  at  the  end  of  25  yearn. 

Showing  the  final  repayment  of  the  loan,  by  the  operation  of 
the  increased  annual  instalment  of  £762'490. 


Statement  XX.  B, 


Rate  of  accumulation,  3  per  cent. 


fear. 

1 

Amount  in 

the  fund 

at  beginning 

of  year. 

Income 

received 

from 

investments. 

Income 
received  from 
Annual        investments  made 
sinking  fund      after  12th  year 
instalment.          3  per  cent. 

Amount  in 

the  fund 

at  end 

of  year. 

Year. 
1 

2 

1 
2 

^ 

3 

3 

4 

The 

amount  in  the  fund  at  the  en 

d  of 

4 

5 

the 

L2th  year. 

£9932-74^ 

;,  is  the  correct 

5 

6 

calci 

dated  amount,  as  shown  by  Calcula- 

6 

7 

tion 

(XY)  2, 

and    by 

the    pro    f 

arm  a 

7 

8 

account,  No.  ] 

,  Chapter 

XV. 

8 

9 

9 

10 

10 

11 

11 

12 

9932-744 

12 

13 

9932-744 

297-984 

762-490 

— 

10993-218 

13 

14 

10993-218 

297-984 

762-490 

31-814 

12085-506 

14 

15 

12085-506 

297-984 

762-490 

64-583 

13210-563 

15 

16 

13210-563 

297-984 

762-490 

98-335 

14369-372 

16 

17 

14369-372 

297-984 

762-490 

133-099 

15562-945 

17 

18 

15562-945 

297-984 

762-490 

168-906 

16792-325 

18 

19 

16792-325 

297-984 

762-490 

205-787 

18058-586 

19 

20 

18058-586 

297-984 

762-490 

243-775 

19362-835 

20 

21 

19362-835 

297-984 

762-490 

282-903 

20706-212 

21 

22 

20706-212 

297-984 

762-490 

323-204 

22089-890 

22 

23 

22089-890 

297-984 

762-490 

364-714 

23515-078 

23 

24 

23515-078 

297-984 

762-490 

407-470 

24983-022 

24 

25 

24983022 

297-984 

762-490 

451-504 

26495-000 

25 

THE  RATES   OF  ACCUMULATION   AND   INCOME        247 


CHAPTER    XXI. 

SINKING  FUND  PEOBLEMS  EELATING  TO  THE 
EATE8  PEE  CENT.  OF  INCOME  AND  ACCUMULA- 
TION {^Continued). 

Variation  C,  in  which  there  is  a  variation  in  the  rate  of 
accumulation  and  also  in  the  rate  of  income  upon  the 
present  investments,  but  without  any  variation  in  the 
period  of  repayment. 

Summary  of  the  methods  of  adjustment.  The  deductive 
METHOD.     Comparison  of  results  obtained  in  this  and 

PREVIOUS       chapters       IN       ALL       PROBLEMS       INVOLVING       A 
VARIATION    IN   THE    RATE    PER    CENT.  STATEMENT    SHOWING 

THE   FINAL    REPAYMENT    OF  THE    LOAN    BY    THE    OPERATION    OF 
THE  AMENDED  ANNUAL  INSTALMENT. 


Summary  of  the  methods  of  adjustment. 

(/)  The  deductive  method,  as  suniviarised  at  the  head  of 
Chapter  XIX,  will  apply,  but  has  been  slightly  viodified  in 
Statement  XXI.  A. 

[II)  The  direct  method,  ivithout  calculation,  as  summarised 
at  the  head  of  Chapter  XX,  will  not  apply  to  this  variation. 

{Ill)  The  annual  increment  {balance  of  loan)  method,  as 
summarised  at  the  head  of  Chapter  XXII,  may  be  used,  but 
will  not  be  applied  to  the  example  under  review. 

{IV)  The  annual  increment  {ratio)  method,  as  sumnnarised 
at  the  head  of  Chapter  XXIII ,  may  be  used,  but  will  not  be 
applied  to  the  example  under  review. 

Note.  The  terms  used  in  the  summaries  above  mentioned 
are  fully  explained  at  the  head  of  Chapter  XXJI .  In  all  the 
above  methods  it  is  imperative  that  the  rates  of  accumulation 
and  of  income  from,  investments  be  uniform  during  the  whole 
of  the  unexpired  or  substituted  period,  of  repayment. 


248         REPAYMENT   OF   LOCAL   AND    OTHER    LOANS 

The  enquiry  into  the  methods  of  adjusting  the  annual 
sinking  fund  instalment  in  consequence  of  any  variation  in 
the  rate  per  cent,  is  now  almost  completed.  Variation  A,  which 
alfected  the  rate  of  accumulation  only,  is  fully  discussed  in 
Chapter  XIX.  In  the  case  of  Variation  B  (Chapter  XX)  the 
varying  factor  is  the  rate  per  cent,  of  income  to  be  yielded 
by  the  present  investments  representing  the  fund.  The  enquiry 
will  noAv  be  completed  by  examining  Variation  C,  in  which 
there  is  a  simultaneous  change  in  both  the  above  rates,  and  the 
deductive  method  will  again  be  used,  as  fully  described  in 
Chapter  XIX,  and  of  which  a  summary  of  the  various  stages  is 
placed  at  the  beginning  of  that  chapter.  The  two  preceding 
chapters  deal  exhaustively  with  all  general  questions  affecting 
the  enquiry,  and  they  will  apply  equally  to  the  present 
variation. 

The  position  of  the  fund  at  the  time  of  making  the  adjust- 
ment is  fully  set  out  in  the  following  Statement  XXI.  A.,  which 
is  similar  to  those  prepared  to  illustrate  the  variations  already 
considered.  These  conditions  are  based  upon  those  obtaining 
when  the  original  calculation  was  made,  and  although  the 
present  example  Avill  be  compared  with  the  conditions  in 
Variation  A  (rate  of  accumulation)  they  will  also  be  compared 
with  those  originally  existing.  As  in  previous  instances  it  will 
be  assumed  that  all  sums  are  added  to  the  fund  and  accumulated 
annually.  In  the  following  chapter  (XXII)  the  whole  of  the 
results  of  the  enquiry  into  the  rate  per  cent,  will  be  compared 
in  order  to  show  the  general  elfect  of  such  rate  upon  the 
accumulation  of  a  sinking  fund.  The  investigation  will  then 
be  extended  in  order  to  derive  a  rule  and  formula  by  means 
of  which  the  adjustments  may  be  made  by  the  more  direct 
annual  increment  (ratio)  method. 

Similar  statements  have  been  prepared  as  in  the  previous 
variations,  namely,  XXI.  A.,  showing  the  amended  annual 
instalment  as  ascertained  by  the  deductive  method ;  and 
XXI.  B.,  showing  the  final  repayment  of  the  loan  by  the 
operation  of  the  amended  annual  instalment  so  ascertained,  as 
shown  by  the  pro  forma  account,  No.  9. 

The  results  already  obtained  may  now  be  briefly  stated, 
both  with  regard  to  the  original  conditions  as  well  as  Variations 
A,  B,  and  C,  in  order  to  show  the  progressive  variations  in  the 
examples  which  will  be  used  later  in  discussing  the  derivation 
of  a  rule  and  formula  which  may  be  ap])liod  to  any  jiroblem 
relating  to  the  rate  per  cent. 


THE  RATES   OF  ACCUMULATION   AND   INCOME        249 

The  siuking-  fuud  iustalment  as  origiually  calculated,  at  a 
rate  of  accumulatiou  of  '6j   per  ceut.,   was 

Calculation  (XV)  1       £680-2o4 

In  Variation  A  (rate  of  accumulation  only),  as 
compared  with  the  original  conditions,  the  rate 
of  income  from  investments  remained  un- 
altered, namely,  3^  per  cent.,  but  the  rate  of 
accumulation  was  reduced  from  -i^  to  o  per 
per  cent.,  requiring  an  additional  instalment, 
as  shown  by  Calculation  (XIX)  :J  of     £32-592 

Amended  instalment  (A),  Statement  XIX.  A.       £T12'826 

In  \'ariation  13  (rate  of  income  only)  as  compared 
with  Variation  A,  the  rate  of  accumulation 
remained  at  3  per  cent.,  but  the  rate  of  income 
from  investments  was  reduced  from  3^  to 
3  per  cent.,  requiring  an  additional  instalment 
as  shown  by  Calculation  (XX)  2  of £49-664 

Amended  instalment  (B),   Statement  XX.  A.       £762490 

In  Variation  C,  as  compared  with  A'ariation  A,  the 
rate  of  accumulation  was  further  reduced  from 
3  to  2^  per  cent.,  and  the  rate  of  income  from 
investments  was  also  reduced  from  3^  to  3  per 
cent.  This  required  an  additional  instalment, 
as  shown  by  Calculation  (XXI)  4,  of  £83-099 
but  part  of  this  was  due  to  the  reduc- 
tion    in     the     rate     of     income     in 

Variation  B,   as  above       £49-664 

£33-435 

Amended  instalment  (C.)      Statement  XXI.  A.        £795925 

which  amended  annual  instalment  is  required  to  be  set 
aside  out  of  revenue  or  rate  and  accumulated  in  addition  to 
the  income  from  the  present  investments,  in  order  to  provide 
the  loan  repayable  at  the  end  of  the  prescribed  period. 

In  Statement  XXI.  A.  following,  the  additional  annual 
instalment  is  ascertained  to  be  £8:5-099.  This  annual  increase 
is  derived  directly  from  the  conditions  in  Variation  A,  and  is 
made  up  of  the  increased  instalment  due  to  the  reduction  in  the 


250    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

rate  oi  income  iu  \  ariation  B,  viz.,  £49  664,  and  the  above 
aiuouut  of  £o6'-i6d  due  to  tiie  variation  in  the  rate  o±  accumula- 
tion. In  the  present  example  there  is  a  variation  in  both  the 
rates  of  income  and  of  accumulation,  and  in  this  respect  it 
combines  the  changes  in  Variations  A  and  13.  In  Variation  13, 
where  there  is  a  change  in  the  rate  of  income  only,  the  annual 
instalment  is  corrected  by  adding  thereto  the  actual  deficiency 
in  the  future  annual  income.  Statement  XXI.  A.  shows  by 
the  deductive  method  that  the  amount  of  original  loan  which 
would  be  unprovided  in  consequence  of  the  concurrent  reduction 
in  the  above  rates  is  £1258"  15,  requiring  an  additional  annual 
instalment  of  £8o"099,  as  shown  by  Calculation  (.XXlj  4.  This 
additional  annual  instalment  is  the  measure  of  the  two  annual 
losses  of  interest,  and  it  is  possible  to  allocate  to  each  rate  the 
proportions  in  which  they  contribute  thereto.  This  will  be  seen 
by  referring  to  the  following  Statement  XXI.  C,  in  which 
column  (2)  contains,  in  the  case  of  Variation  A,  the  original 
and  additional  annual  instalments,  and  also  the  income  from 
investments,  making  up  the  amended  annual  increment.  This 
agrees  with  Statement  XIX.  A.  in  total,  but  the  income  from 
investments  at  3^  per  cent,  has  been  divided  as  between  o  and 
J  per  cent,  iu  order  to  compare  this  variation  with  Variation  C. 
The  final  column  (8)  shows  the  deficiency  of  original  loan 
caused  by  the  accumulation  of  each  of  the  component  parts  of 
the  annual  increment  at  2|  per  cent,  in  Variation  C,  instead 
of  at  3  per  cent.,  as  in  Variation  A. 

This  deficiency  is  arrived  at  by  deducting  the  amount  of 
loan  in  column  (7)  from  the  amount  in  column  (4),  and  is  made 
up  as  follows,  expressed  in  terms  of  original  loan  :  — 

Deficiency  due  to  the  reduction  in  the  rate  of 
accumulation,  from  3  to  2^  per  cent.,  of  items 
1  to  3     £482-51 

Deficiency  due  to  the  reduction  in  the  accumulation 

of  the  decrease  in  income,  item  4 £23'70 


£506-21 


Deficiency   of   annual    income   accumulated   at   2i 

percent £75194 

£1258-15 


THE  RATES   OF  ACCUMULATION   AND   INCOME        251 

The  deficiency  of  <£1258'15  of  loan  requires  a  total  additional 
annual  instalment  of  i>8o'099,  as  previously  ascertained,  which 
is  made  up  of  :  — 

the  loss  of  income  from  investments £49'664 

and  the  loss  owing  to  the  reduction  in  the  rate  of 

accumulation         £;ilj"4y5 


£83-099 


The  above  amount  of  <£33'435  includes  the  loss  of  accumula- 
tion not  only  upon  the  remaining  portion  (£1010'810)  of  the 
present  annual  increment  in  Variation  A,  as  shown  in  column  2, 
but  also  upon  the  reduction  in  the  annual  income,  viz.,  £49664. 
This  proves  that  when,  as  in  Variation  C,  the  reduction  in  the 
rate  of  income  from  investments  is  accompanied  by  a  reduction 
in  the  rate  of  accumulation,  the  additional  annual  instalment  is 
measured,  not  by  the  actual  reduction  in  the  annual  income,  as 
in  Variation  B,  but  by  the  annual  deficiency  of  income  increased 
in  the  ratio  that  the  amount  of  £1  per  annum  at  the  past  rate 
bears  to  the  amount  of  £1  per  annum  at  the  future  rate,  in  each 
case  for  the  same  number  of  years, being  the  unexpired  portion 
of  the  original  repayment  period.  This  will  be  referred  to  later 
in  Chapter  XXII,  when  discussing  Calculation  (XXII)  E.  with 
the  object  of  arriving  at  a  method  of  making  the  adjustment 
by  the  more  direct  annual  increment  (ratio)  method.  In  that 
case  the  comparison  will  be  made  between  Variation  C  and  the 
original  conditions,  but  the  same  principles  apply,  and  the 
above  table  may  be  again  referred  to  with  advantage. 
(Statement  XXI.  C.  follows.) 


252         REPAYMENT   OF   LOCAL   AND   OTHER   LOANvS 


The  Rate  per  cent.  Statement  XXI.  A, 

The  Deductive  Method. 
Variation  C,  rates  of  accumulation  and  income  combined. 

Showing  the  method  of  adjusting  the  annual  instalment  in 
consequence  of  a  variation  in  the  rate  of  accumulation 
and  also  in  the  rate  of  income  upon  the  present  investments, 
but  without  any  variation  in  the  period  of  repayment. 

This  example  is  compared  with  the  original  conditions 
as   modified   by   Variation   A. 

Conditions  before  adjustment  fat  end  of  12th  year) 

Amount  of  loan  repayable  'in  25  years £26,495 

Amount  in  the  fund  (at  the  end  of  12th  year)   ...     £9932-74 

Present  annual  income  (previously)  received  there- 
from, at  3|  per  cent,  per  annum     £347'648 

Present  annual  instalment,  to  be  accumulated  for 

13  years  at  3  per  cent £712'826 

Present  annual  increment £1060474 

Variation  from  the  above  conditions  : — 

The  rate   of  accumulation   of   the   fund   is   reduced   from 
3  to  2|  per  cent. 

The  rate  of  income  yielded  by  the  present  investments  is 
reduced  from  3i  to  3  per  cent. 

Future  annual  income £297984 

The  future  rate  of  accumulation 2^  per  cent. 


THE  RATES   OF  ACCUMULATION   AND   INCOME        253 

Present  investments     (at     end     of     12th     year), 

representing  tlie  amount  now  in  the  fund     ...        £9932' 74 

Future  annual  income  from  present  investments  : — 

Amount   of  an  annuity  of       £29T'984 


accumulated  for  13  years,  at  2|  per  cent. 

Calculation  (XXI)  1       £451101 

Original  annual  instalment : — 

Amount  of  an  annuity  of £680'234 


accumulated  for  13  years,  at  2^  per  cent. 

Calculation  (XXI)  2     £1029904 

Additional  annual  instalment  (Vdriafion  A) :  — 
Amount  of  an  annuity  of       £32'592 


accumulated  for  13  years,  at  2^  per  cent. 

Calculation  (XXI)  3         £493-46 


Provision  already  made  -will  repay  loan  of £25236'85 

Additional  annual  instalment  required  : — 

Balance,  being  amount  of  original  loan  un- 
provided for  owing  to  tlie  above  decrease  in  the 
rate  of  accumulation,  and  in  the  rate  of  income 
from  investments  requiring  an  additional 
annual  instalment,  to  be  set  aside  and 
accumulated  for  13  years  at  2|  per  cent £125815 

Additional  annual  instalment 

Calculation  (XXI)  4  £83-099 


Amount  of  orio-inal  loan    £26495-00 


Amended  annual  increment,  J)ciuf/  :  — 

Income  from  investments         £297-984 

Amended  annual  instalment 795-925 


£1093-909 


254 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


The  Rate  per  cent. 


Statement  XXI.  B. 


Variation  C,  rates  of  accumulation  and  income 

Showing  the  final  repayment  of  the  loan,  by  the  operation  of 
the  sinking  fund  after  making  the  adjustment  in  the 
annual  instalment,  consequent  upon  a  variation  in  the  rate 
of  accumulation,  and  also  in  the  rate  of  income  upon  the 
present  investments,  but  without  any  variation  in  the 
period  of  repayment. 


Present  investments    (at  end  of  12th  year) 


Equivalent 

amount  of 

original  loan. 

£9932-74 


Amended  annual  increment : — 

Original  annual  instalment     £680'234 

Additional.  Variation  A        32-592 

ditto.  Variation  C        83-099 


Total  out  of  revenue  ... 
Income  from  investments 


.      795-925 
.      297-984 

£1093-909 


Amount  thereof,  accumulated  for  13  years  at 

2i  per  cent.  Calculation  (XXI)  5     £1656226 


Amount  of  original  loan 


£2649500 


Amended  annual  instalment    ...  £795925 


THE  RATES   OF  ACCUMULATION   AND   INCOME        255 


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256 


REPAYMENT   OF   LOCAL   AND   OTHETl   LOANvS 


Pro  forma  Sinking  Fund  Account,  No.  9. 

A  Yariatiou  in  the  rate  of  Accumulation,   as  well  as  in   the 
Eate  of  Income  upon  the  present  Investments. 

Loan   of  £26,495,  repayable   at  tJte   end  of  25  years. 

Showing  the  final  eepaymext  of  the  loan,  by  the  operation  of 
the  increased  annual  instalment  of  £795"  925. 


Statement  XXI.  B. 


Eate  of  Accumulation,  2^  per  cent. 


Year. 

Amount  in 

the  fund 

at  beginning 

of  year. 

Income 

received 

from 

investments. 

Annual 

Sinking 

Fund 

instalment. 

Income 

received  from 

investments 

'2i  per  cent. 

Amount  in 

tlie  fund 

at  end 

of  year. 

Vea 

1 

1 

2 

2 

3. 

3 

4 

The 

amount  in  the  fund 

at  the  end  of 

4 

5 

the 

12th  year, 

£9932-744 

,  is  the  correct 

5 

6 

calculated  amount,  as  shown  by  Ca 

Icula- 

6 

7 

tion 

(XY)  2, 

and    by    1 

he    pro 

forma 

7 

8 

account,  No.   ] 

,   Chapter 

XT. 

8 

9 

9 

10 

10 

11 

11 

12 

9932-744 

12 

13 

9932-744 

297-984 

795-925 

— 

11026-653 

13 

14 

11026-653 

297-984 

795-925 

27-348 

12147-910 

14 

15 

12147-910 

297-984 

795-925 

55-379 

13297-198 

15 

16 

13297-198 

297-984 

795-925 

84-111 

14475-218 

16 

17 

14475-218 

297-984 

795-925 

113-562 

15682-689 

17 

18 

15682-689 

297-984 

795-925 

143-749 

16920-347 

18 

19 

16920-347 

297-984 

795-925 

174-690 

18188-946 

19 

20 

18188-946 

297-984 

795-925 

206-405 

19489-260 

20 

21 

19489-260 

297-984 

795-925 

238-913 

20822-082 

21 

22 

20822-082 

297-984 

795-925 

272-233 

22188-224 

22 

23 

22188-224 

297-984 

795-925 

306-387 

23588-520 

23 

24 

23588-520 

297-984 

795-925 

341-394 

25023-823 

24 

25 

25023-823 

297-984 

795-925 

377-268 

26495000 

25 

Section  IV. 
Sinking    Fund   Problems. 


The   Annual   Increment. 


259 


CHAPTER  XXII. 

THE  ANNUAL  INCREMENT  METHODS. 

Definition  of  terms  relating  to  the  annual  increment  and 
the    methods    of    ascertaining    the    amended    annual 

INSTALMENT       BASED       THEREON.  ThIS       APPLIES       TO       ALL 

VARIATIONS  IN  THE  RATE  OF  ACCUMULATION  AND  THE  PERIOD 
OF  REPAYMENT,  WITH  OR  WITHOUT  ANY  VARIATION  IN  THE 
RATE  OF  INCOME  UPON  THE  PRESENT  INVESTMENTS  REPRE- 
SENTING THE  FUND. 

SINKING  FUND   PROBLEMS   RELATING   TO   THE 
RATE  PER  CENT.   OF  ACCUMULATION. 

Methods  of  ascertaining  the  amended  annual  instalment  by 
direct  calculation  in  terms  of  the  annual  increment. 
Comparison  of  the  results  already  obtained  in 
Chapters  XIX,  XX,  and  XXI  in  terms  of  the  annual 
instalment  with  those  obtained  by  means  of  the  annual 
increment  and  the  varying  rates  of  accumulation. 
The  annual  increment  (balance  of  loan)  method. 


Summary  of  the  methods  of  adjustment. 

(Z)  The  deductive  method,  as  siimmarised  at  the  head  of 
Chapter  XIX, 

as  to  the  rate  of  acctnmdation,  Statement  XIX.  A. 

as  to  the  rate  of  income  and  the  rate  of  accumulation^ 

Statement  XXI.  A. 

(II)  The  direct  method,  tcithout  calculation,  as  s^immarised 
at  the  head  of  Chapter  XX,  will  not  apply  to  these  variattons. 

{Ill)   The  annual  increment  {balance  of   loan)   method,    as 
summarised  heloiv,  is  illustrated  in  the  text. 

(IV)   I  he  annual  increment  (ratio)  viethod,  as  summarised 
at  the  head  of  ( 'liaptvr  XXIII,  State mcnt  XXII.  C. 


26o    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Note.  In  all  cases  where  the  adjustment  is  made  by  the 
annual  increment  methods  it  is  imperative  that  the  rates 
per  cent.,  both  of  accumulation  and  income  from  investments, 
he  uniform^  during  the  whole  of  the  unexpired  or  substituted 
period  of  repayment. 

The  Annual  Increment  Methods,  Definition  of  Terms. 

The  present  annual  increment,  at  the  time  of  making  an 
adjustment  in  the  annual  instalment,  consequent  upon  a 
variation  in  the  rate  of  accumulation,  or  in  the  period  of 
repayinent,  or  in  both  these  factors  combined^  is  composed  of : 

1.  The  present  or  original  anmial  instalment,  tvhich  has  been 

set  aside  and  added  to  the  sinking  fund  up  to  the  time 
of  making  the  adjustment,  and 

2.  The  present  anmial  income  from  investments,  representing 

the  fund,  whicli  has  been  received  up  to  the  date  of 
making  the  adjustment. 

The  future  or  amended  annual  increment,  consequent  upon 
a  variation  in  either  or  both  of  the  factors  of  rate  per  cent,  of 
accumulation  and  period  of  repayment,  is  composed  of:  — 

1.  The  future   or   amended   anmial   instalment^    required   to 

be  set  aside  and  added  to  the  sinking  fund  in  consequence 
of  the  above  variation  or  variations,   and 

2.  The  future  annual  income  from  investments,  representing 

the  fund  at  the  date  of  making  the  adjustment  whether 
the  rate  of  income  ^ipon  such  investments  remains  un- 
altered, or  is  varied. 

The  annual  increments,  as  above  described^  are  the  primary 
and  final  factors  in  all  the  adjustments  by  these  methods. 

The  past  rate  denotes  the  rate  of  accumulation  upon  which  is 
based  the  present  or  original  annual  instalment  included 
in  the  present  annual  increment. 

The  future  rate,  denotes  the  rate  of  accumulation  to  be  tised 
instead  of  the  past  rate  to  calculate  the  future  or  amended 
annual  increment.  It  will  he  ihe  same  as  the  past  rate 
in  problems  involving  a  variation  in  the  period  of 
repayment  only,  without  any  variation  in  the  rate  of 
accuTnulation. 


THE    RATE    OF    ACCUMULATION  261 

The  uriexpired  ■period,  denotes  the  unexpired  portion,  at  the 
time  of  making  the  adjustment^  of  the  original  repay- 
ment period  upon  irhich  the  present  or  original  annual 
instalment  was   based. 

The  substituted  period,  denotes  the  increased  or  reduced  number 
of  years  over  ivJtich  the  future  or  amended  annual 
instalments  shall  be  spread  and  at  the  end  of  which  the 
full  amount  of  the  loan  will  be  repayable.  It  will  be 
the  same  as  the  unexpired  period  in  problems  involving 
a  variation  in  the  rate  of  accumulatio7i  only,  without 
a  variation  in  the  period  of  repayment. 

The  income  from  investments ^  representing  the  amount  in  the 
fund  does  not  enter  into  the  actual  calculation  except  as 
a  component  part  of  the  present  and  future  or  amended 
annual  increments,  as  above  defined. 

The  future  or  amended  annual  instalment,  is  obtained  in  all 
cases  by  deducting,  from  the  ascertained  amended  annual 
increment,  the  future  annual  income  from  the  present 
investments  representing  the  fund,  u-hether  the  rate  of 
income  upon  such  investments  remains  unaltered  or  is 
varied. 

Note.  The  foregoing  definitions  will  be  referred  to  in 
subsequent  chapters,  loithout  any  further  explanation  or 
amplification. 

Geneeal  summaey  of  the  annual  increment  (balance  of 
loan)  method,  of  ascertaining  the  amended  annual  sinking 
fund  instalment  due  to  a  variation  in  either  the  rate  of 
accumulation,  the  period  of  repayment,  the  rate  of  income  upon 
the  present  investments  representing  the  fund,  or  any  of  these 
factors  in  combination.  The  terms  used  in  the  following 
summary  are  fully  explained  above. 

{!)  Ascertain  the  value  of  the  present  investments  in  the 
manner  already  described^  and  deduct  the  value  so 
obtained  from  the  amount  of  the  original  loan. 

[2)  The  remainder  represents  the  balance  of  loan  to  be 
provided  by  the  accumulation  of  the  future  or  amended 
annual  increment,  as  previously  defined,  for  the  un- 
expired or  substituted  repayment  period  at  the  future 
rate   of  accumulation. 


262         REPAYMENT   OF   LOCAL   AND   OTHER   LOANvS 

(J)  Calculate  tJte  annuifij,  or  annual  increnicnt^  to  be  added 
to  the  fund  and  accumulated  for  the  period  and  at  tJic 
rate  per  cent,  as  in  {2). 

[4)  From  the  amended  anntuil  increment  ascertained  as  in  (3), 
deduct  the  future  annual  income  to  he  received  from  the 
present  investments  during  the  tvhole  of  the  unexpired 
or  substituted  repayment  period. 

(J)  'The  remainder  ivill  represent  tJie  future  or  amended 
annual  instalment  to  be  charged  to  revenue  or  rate 
account,  and  added  to  the  fund,  during  the  whole  of  the 
unexpired  or  substituted  repayment  period. 

{6)  Prepare  a  statement  showing  the  final  repayment  of  the 
loan  by  the  operation  of  the  sinking  fund  under  the 
amended  conditions. 

(7)  Prepare  a  pro  forma  account  showing  the  amount  lohich 
should  be  in  the  fund  at  the  end  of  each  year  of  the 
unexpired  or  substituted  repayment  period. 

Memo.  In  tJie  event  of  the  income  from  investments  not 
being  uniform  over  the  tvhole  of  the  repayment  period^  proceed 
by  the  method  in  Chapter  XXYII. 


Sinking  Fund  Problems,  relating  to  the  rate  per  cent. 
OF  accumulation.  The  Annual  Increment.  Iu  previous 
chapters  dealing  with  the  three  variations  in  the  rates  per  cent, 
of  accumuhition  and  income  from  the  present  investments, 
the  amended  annual  instalment  has  been  ascertained  by  the 
deductive  method  described  in  Chapter  XIX. 

This  method  is  based  upon  (1)  the  value  of  the  present 
investments  representing  the  fund  as  described  in  Chapter  XI^  ; 
(2)  the  annual  income  to  be  received  therefrom,  and  (3)  the 
original  annual  instalment.  All  these  factors  have  been 
reduced  to  equivalent  amounts  of  original  loan  ultimately 
repayable,  in  order  to  ascertain  the  deficiency  in  the  fund  at 
the  end  of  the  repayment  period  due  to  the  reduction  in  the 
rate  per  cent,  of  income  or  of  accumulation. 

This  deficiency  of  original  loan  ultimately  repayable  has 
been  converted  into  an  equal  annual  sinking  fund  instalment, 
to  be  provided  out  of  revenue  or  rate,  in  addition  to  the  original 
instalment.  A  statement  has  been  prepared  showing  in  each 
case  the  final  re})aymeni  of  the  loan  by  the  operation  of  the 
amended  annual  instalment  so  ascertained  at  the  end  of  the 
12th  year. 


THE    RATE    OF    ACCUMULATION  263 

In  these  statements  tlie  amount  of  the  loan  has  been  divided 
into  two  parts;  the  first  (£9932-74)  being  the  value  of  the 
present  investments  representing  the  fund,  and  the  second 
(£16562-26)  being  the  amount  of  loan  to  be  provided  at  the  end 
of  the  repayment  period  by  the  accumulation  of  the  future  or 
amended  annual  increment,  which  consists  of  :  — 

1.  Income  from  the  present  investments. 

2.  The  original  annual  instalment. 

3.  The  additional  annual  instalment  ascertained  in  the  above 

manner, 
thereby  proving  the  accuracy  of  the  results  obtained  by  the 
deductive  method.     But  the  original  annual  instalment  is  the 
only  constant  factor,  although  it  may  in  future  accumulate  at  a 
lower  rate   than  was   originally   estimated.     Consequently,    in 
arriving    at    the    future    or    amended    annual    instalment    two 
variable  factors  have  to  be  considered,  namely,  (1)  the  rate  of 
income    upon    the    present    investments,    and    (2)    the    rate   of 
accumulation.     These  two  factors  of  rate  per  cent,   are  most 
important  in  the  after  consideration  of  the  problem  because 
they  may  vary  in  different  directions  and  are  not  in  any  way 
related.     But  any  difficulty  may  be  eliminated  by  treating  the 
amount  of  the  future  annual  income  to  be  received  from  the 
present  investments,  at  the  future  rate  per  cent,  of  income,  as 
an  annuity  certain  in  the  same  manner  as  the  original  annual 
instalment.     These  two  factors  together  constitute  the  future 
or  amended  annual  increment  of  the  fund,  which  is  acted  upon 
by   the    future    rate    of    accumulation    only,    consequently    the 
problem  has  been  reduced  to  an  annuity  certain  for  a  definite 
term  at  a  given  rate  per  cent.     The  annual  income  from  the 
present  investments,  included  in  the  present  annual  increment 
in  all  adjustments  made  by  this  method,  is  the  annual  amount 
which  has  been  received  in  the  past  and  is  not  the  future  annual 
income    which    will    be    yielded    during    the     unexpired     or 
substituted  period  of  repayment.     This  is   one  of  the   funda- 
mental principles  of  the  annual  increment  (ratio)  method.     The 
enquiry  is  thereby  transferred  from  the  annual  instalment  to 
the  annual  increment,  and  as  this  is  an  annuity  of  fixed  amount 
it  is  possible  to  arrive  at  a  formula,  and  a  rule  based  thereon. 
The  annual  increment  has  been  fully  described  in  Chapter  XIV. 
In  Chapters  XIX,   XX,  and  XXI,  three  variations  in  the 
rate  per  cent,  have  been  considered,  and  the  amended  annual 
instalment  in  each  case  has  been  ascertained  by  the  deductive 
method.     Up  to  this  point  the  examples  have  been  considered 
only  as  individual  problems,  but  they  will  now  be  treated  in 


264    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

combination.  In  order,  however,  to  avoid  undue  reference  to 
previous  chapters,  the  following  Statement  XXII.  A.  has  been 
prepared  containing  the  whole  of  the  conditions  in  each  case 
and  the  actual  results  previously  obtained. 

A  further  classified  Statement  XXII.  B.  has  been  prepared 
showing  the  initial  conditions  in  each  case  and  giving 
references  to  methods  and  calculations  by  which  the  results 
have  been  obtained.  It  should  again  be  mentioned  that, 
although  in  each  A-ariation  a  reduction  has  been  assumed  in  the 
rates  per  cent.,  as  being  more  likely  to  occur  in  practice,  yet 
the  same  principles  and  methods  will  apply  equally  to  an 
increase  in  both  rates  per  cent.,  or  to  an  increase  in  one  rate 
and  a  reduction  in  the  other. 

Statement  XXII.  A.  (page  265)  contains  full  details  of  the 
amended  annual  instalments  found  by  the  deductive  method  in 
the  three  Variations  A,  B,  and  C,  which  are  derived  one  from 
the  other  and  from  the  original  conditions  by  gradual  variations 
in  the  rates  of  income  and  accumulation.  There  is  therefore  a 
definite  relation  between  the  original  annual  increment  of 
<£1027'882  and  the  successive  annual  increments  in  Variations 
A,  B,  and  C,  leading  to  the  final  annual  increment  of  £1093909 
in  Variation  C.  This  relation  depends  upon  the  respective 
rates  of  accumulation  in  the  four  examples,  and  by  this  means 
it  is  possible  to  derive  the  rule  and  formula  required.  State- 
ment XXII.  B.  contains  the  annual  increments  only,  and  shows 
the  rates  per  cent,  of  income  and  accumulation  in  each  case. 
All  these  annual  sums  are  derived  from  a  common  source,  and 
therefore  may  be  treated  as  simple  annuities  for  a  term  without 
reference  to  any  principal  sum  or  other  factor  than  the  rate  of 
accumulation.  In  Statement  XXII.  B.  the  variations  in  the 
rate  per  cent,  are  divided  into  two  classes  depending  upon  the 
rate  of  accumulation.  The  first  class  contains  the  problems 
in  which  the  rate  of  accumulation  remains  unaltered,  and  there 
is  not  therefore  any  necessity  to  sub-divide  the  class  as  regards 
any  variation  in  the  rate  of  income  on  the  investments,  because, 
as  ascertained  in  considering  Variation  B,  there  is  not  any 
question  of  compound  interest  involved.  It  is  only  necessary 
to  correct  the  original  annual  instalment  by  adding  to  or 
deducting  therefrom  the  difference  between  the  annual  amounts 
of  income  yielded  by  the  present  investiuents  at  the  past  and 
future  rates  respectively.  The  second  class  includes  cases  in 
which  there  is  a  variation  in  the  rate  of  accumulation,  and  this 
class  may  be  sub-divided  according  as  the  rate  of  income  upon 
the  present  investments  is  unaltered  or  is  varied.     Although  it 


THE    RATE    OF    ACCUMULATION  265 

will  be  found  that  both  sub-divisious  of  this  class  may  be 
treated  by  one  and  the  same  rule  and  formula,  the  present 
distinction  is  useful  in  giving-  emphasis  to  the  fact.  It  will 
also  be  seen,  in  dealing  with  problems  in  which  there  is  a 
change  in  the  rate  of  income  upon  investments  as  well  as  in  the 
rate  of  accumulation,  Chapter  XXI,  that  the  reason  why  the 
rule  applies  is  not  so  obvious  as  in  the  case  of  a  simple  variation 
in  the  rate  of  accumulation  only. 

Class  I.       Variations  in  the  rate  of  incoine  from  investments 

only,      the      rate      of     accumulation     remaining 

unaltered. 

Problems  of  this  nature,  in  which  the  variation  affects  only 

the  rate  of  income  on  the  present  investments,  but  in  which  the 

rate    of    accumulation    remains    the    same,    have    been    fully 

described  in  Chapter  XX,  Variation  B.     The  calculation  of  the 

amended  annual  instalment  in  such  cases  may  be  made  by  the 

deductive  method,  Statement  XX.  A.,  which  applies  equally  to 

all  manner  of  variations  in  the  rate  per  cent.     But  Statement 

XX.   C.   shows  that  the   amended   annual   instalment  may  be 

arrived   at   by   a    simple    direct    method,    without    calculation, 

although   the   deductive    method    may    be    used   to    prove    the 

accuracy  of  the  conclusions. 

Class  II.      Variations  in  the  rate  of  accumulation. 

This  class  has  been  sub-divided  into  two  groups,  as  shown 
in  Statement  XXII,  B.  as  follows:  — 

{A)  In  which  the  rate  of   income   upon  the  present  invest- 
ments is  unaltered. 
(B)  In  which  the  rate  of  income  upon  the  present  investments 
is  varied. 
Each   of  these   sub-divisions  will   be   considered   in   detail, 
taking  as  examples  the  figures  given  in  Statement  XXII.  B. 

The  Rate  per  cent.  Statement  XXII,  A. 

Variation  A.     Hate  of  accumulation  only.  Chapter  XIX. 

Variation  B.     Rate  of  income  only.  Chapter  XX, 

Variation  C.      Rates  of  accumulation  and  income  combined.    Chapter  XXI. 

Showing,  at  the  end  of  the  12th  year,  under  the  original  conditions,  and 
under  each  variation  :  — 

(1)  The  present,   and  future  or  amended  annual  increments. 

(2)  The  additional  annual  instalment  distinguishing  between  the 

loss  of  income  from  the  present  investments,  and  the  reduction 
in  the  rate  of  accumulation. 

(3)  The  provision  of  the  future  annual  increment  from  internal 

and  external  sources. 


266    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Loan  ;^26,495.     Amount  in  the  fund  at  end  of  Original  Variation  Variation          Variation 

12th  year,  ;{;», 03-2 -74.  Conditions  A  B                       C 

Future  rate  of  accumulation 3^  3  3              2j 

Future  yield  on  present  investments  3|  3|  3              3 
\.  Present  annual  increment :  — 

Original  instalment     680234  080-234  680234     680-234 

Income  from  present  invest- 
ments at  end  of  12tli  year, 

at  above  rates 347-648  347-648  297-984     297-984 


Present  annxial  increment 
wliicb  will  continue  to  be 
accumulated  at  reduced  rate 
of  accumulation 1027-882  1027-882     978-218     978-218 

Additional  annual  instahnents 
to  make  good  tbe  loss  of  in- 
terest on  present  investments 
and  future  accumulations,  to 
be  added  to  tbe  original 
annual  instalments  and  pro- 
vided out  of  revenue  or  rate         Nil  32-592       82-256     115-691 


Future  annual  increment  1027-882  1060474  1060-474  1093909 

II.  Tbe  above  additional  annual 
instalments,  as  compared  witli 
tbe  original  conditions  are 
made  up  as  follows  :  — 

1,  Decrease  in  income  from  tbe 

present  investments  ...         ]S'il  Xil  49-664       49'664 

2.  Decrease      in     interest      on 

future    accumulations    due 

to    reduction    in    rate     of 

accumulation  :  — 

Variation  A       —  32592       32-592       32-592 

A'ariation  C        —  —  —  33-435 


Nil         32-592      82-256     115-691 


III.  Future  annual  increment  to  he 
provided  as  follows  :  — 
A.  To  be  taken  out  of  revenue  or 

rate  :  — 
Original    annual    instalment     680234     680-234     680-234     680-234 
Deficiency  in  future  income 

from   present    investments         —  —  49-664       49664 

Additional  annual  instalment 

to  compensate  for  decrease 

in  rate  of  accumulation ...         —  32-592       32-592       66027 


680-234     712-826     762490     795925 
B.  Income     to     be     received     in 
future    from   present    invest- 
ments          347-648     347-648     297-984     297-984 


Future  annual  increment  1027882  1060-474  1060-474  1093-909 


X 


THE    RATE    OF    ACCUMULATION  267 


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268    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Class  II  (A).  In  which  the  rate  of  aceuniulation  is  varied,  hut 
in  ^vhich  the  rate  of  income  upon  the  present 
investments  is  unaltered. 

TJie  following  examples  in  Statement  XXII.  B.  fall  under 
this  head  :  — 

(Ij  In  Variation  A,  as  compared  with  the  original  conditions, 
the  rate  of  income  is  in  each  case  3|  per  cent.,  but  the 
rate  of  accumulation  is  reduced  from  3|  to  3  per  cent. 
The  effect  as  ascertained  by  the  deductive  method  (State- 
ment XIX.  A.)  is  to  increase  the  annual  increment  from 
£1027-882  to  £1060-474.  See  also  Calculation  XXII.  C, 
where  the  same  result  is  obtained  by  the  annual 
increment  (ratio)  method. 

(2)  In  Variation  C,  as  compared  with  Variation  B,  the  rate 
of  income  is  in  each  case  3  per  cent.,  but  the  rate  of 
accumulation  in  Variation  C  is  reduced  from  3  to  2^  per 
cent.  The  effect,  as  will  be  seen  from  Statement  XXII. 
B,,  is  to  increase  the  annual  increment  from  £1060474 
to  £1093-909.  This  adjustment  is  worked  out  in  detail 
in  Calculation  XXII.  D. 

Chapter  XIX  deals  very  fully  with  the  process  of  finding, 
by  the  deductive  method,  the  amended  annual  instalment 
consequent  upon  a  variation  in  the  rate  of  accumulation  only, 
taking  as  an  example  the  original  conditions  as  modified  by 
Variation  A.  In  both  cases  the  present  annual  increment 
consists  of  : — - 

Income  from  investments     £347-648 

Original  annual  instalment 680-234 


£1027-882 


but  under  the  original  conditions  this  annual  increment 
accumulated  at  3|  per  cent.,  whereas  in  Variation  A  the  rate  of 
accumulation  Avas  reduced  to  3  per  cent.  This  requires  an 
additional  annual  instabnent  of  £32"592  to  be  set  aside  out 
of  revenue  or  rate,  as  found  by  Calculation  (XIX)  3.  But  it  is 
apparent  that  this  represents  the  deficiency  in  the  accumulation 
of  £1027-882  per  annum  at  3  per  cent,  instead  of  at  3^  per  cent., 
and  is  measured,  not  by  the  actual  ratio  between  3  and  3|  per 
cent.,  but  by  tlic  ratio  (ixisting  between  the  respective  amounts 
of  £1  per  annum   for  13  years  at  those  rates.     These  amounts 


THE    RATE    OF    ACCUMULATION  269 

are  given  in  Table  III  in  the  published  tables  of  compound 

interest. 

The  same  principle  applies  to  Variation  C,  as  compared  with 
Variation    B,    as   will   be    seen    by    the    following    Calculation 

XXII.  D. 

The  remarks  upon  Calculation  XXII.  C.  relating  to  Varia- 
tion A,  as  compared  with  the  original  conditions,  apply  equally 
to  this  case. 

Class  II  (B).     In  which  the  rate  of  accumulation  and  the  rate 
of  income  upon  the  present  investments  are  both 
varied. 
The  following  examples  in  Statement  XXII.  B.  fall  under 

this  head  :  — 

(1)  In  Variation  C,  as  compared  with  the  original  conditions, 

the  rate  of  income  is  reduced  from  o^  to  3  per  cent., 
and  the  rate  of  accumulation  is  reduced  from  3^  to  21  per 
cent.  The  effect,  as  found  by  Calculation  XXII.  E.,  is 
to  increase  the  annual  increment  from  £1027-882  to 
£1093-909. 

(2)  In  Variation  C,  as  compared  with  Variation  A,  the  rate 

of  income  is  reduced  from  3i  to  3  per  cent.,  and  the  rate 
of  accumulation  is  reduced  from  3  to  2^  per  cent.     The 
effect,  as  will  be  seen  from  Statement  XXII.  B.,  is  to 
increase    the     annual     increment     from     £1060-474     to 
£1093-909.     This  calculation  is  not  worked  out  in  detail, 
but  follows  from  the  premises  as  a  matter  of  course. 
(3)  In  Variation  B,  as  compared  with  the  original  conditions, 
the  rate  of  income  is  reduced  from  3i  to  3  per  cent.,  and 
the  rate  of  accumulation  is  reduced  from  ^  to  3  per  cent. 
The  effect,   as  will  be  seen  from   Statement   XXII.  B., 
is  to  increase  the  annual  increment  from  £1027-882  to 
£1060-474.     This  calculation,  also,  is  not  worked  out  m 
detail. 

In  Calculation  (XXII)  E  the  original  conditions  are  com- 
pared with  Variation  C  in  which  there  is  a  reduction  in  the  rate 
of  income  from  present  investments  from  3^  to  3  per  cent.,  or 
£49-664  per  annum;  and  at  the  same  time  a  reduction  in  the 
accumulation  rate  from  3i  to  2i  per  cent.  Although  it  is  not 
so  obvious  as  in  those  cases  where  there  is  a  variation  m  the 
rate  of  accumulation  only,  yet  the  same  rule  applies,  as  will  be 
seen    by    the    following    considerations.       The    method    is    a 


270    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

combination  of  those  previously  considered.  It  resembles 
Variation  A  in  that  the  rate  of  accumulation  is  reduced,  and 
it  is  therefore  necessary  to  increase  the  original  annual 
increment  in  proportion  to  the  respective  amounts  of  £1  per 
annum  for  13  years  at  the  past  and  future  rates  of  accumulation, 
as  in  Variation  A.,  Calculation  XXII.  C.  It  resembles 
Variation  B  only  to  the  extent  that  the  actual  decrease  in  the 
annual  increment  due  to  the  reduced  annual  income  of  £49'664 
must  be  added  to  the  fund  as  an  additional  annual  instalment  to 
be  provided  out  of  revenue  or  rate.  But  in  Variation  B  there 
is  not  any  reduction  in  the  rate  of  accumulation,  as  in  this 
case;  and  in  Variation  B,  therefore,  the  annual  loss  of  income 
on  the  present  investments  is  the  actual  measure  of  the  deficiency 
in  tlie  annual  instalment.  In  this  case  there  is  a  reduction 
in  the  future  income  from  the  present  investments  accompanied 
by,  and  acted  upon  by,  a  reduction  in  the  rate  of  accumulation; 
consequently  if  the  original  annual  instalment  be  increased  by 
the  loss  of  income  only,  as  in  Variation  B,  the  fund  will  lose 
the  accumulation  on  that  sum  due  to  the  reduction  in  the  rate 
of  accumulation.  It  is  clear,  therefore,  that  the  original  annual 
instalment  must  be  increased,  not  by  the  actual  loss  of  income, 
of  ^  per  cent.,  or  £49'664,  which  under  the  original  conditions 
accumulated  at  3^  per  cent.,  but  by  a  larger  annual  amount 
which,  accumulated  at  2^  per  cent,  only,  will  at  the  end  of  the 
repayment  period  amount  to  the  same  sum.  This  question  has 
been  very  fully  discussed  in  Chapter  XXI,  dealing  with 
Variation  C,  which  contains  a  useful  Statement  XXI.  C, 
which  may  be  consulted  in  this  connection.  Therefore,  the 
actual  rate  per  cent,  of  income  from  the  present  investments 
does  not  enter  into  the  calculation  of  the  annual  increment, 
which,  as  shown  in  Statement  XXII.  E.,  is  exactly  similar  in 
principle  to  Calculations  XXII.  C.  and  XXII.  D. 

On  comparing  the  whole  of  the  above  Calculations  XXII. 
C'.,  D.  and  E.,  it  will  be  seen  from  the  formulae  at  the  heading 
of  each  that  they  all  follow  the  same  rule,  although  the 
conditions  in  each  are  diiferent.  It  may  appear  s\;perfluous  to 
include  them  all,  but  they  will  be  referred  to  again  in  order  to 
illustrate  the  variations  in  (1)  the  period  of  repayment,  in 
Chapter  XXIV,  and  (2)  the  period  of  rcjiayment  accompanied 
bv  a  variation  in  the  rate  per  cent,  of  accumulation,  in  Chapter 
XXVI. 

The  Axnual  Increment  (balance  of  loan)  Method.  Three 
adjustments  have  been  made  relating  to  variations  in  the  rate 
per  cent,  of  accumulation  as  follows:  — 


THE    RATE    OF    ACCUMULATION  271 

Class  II  (A).     Variation  A  from  the  original  conditions, 

Calculation  XXII.  C. 
ditto.  Variation  C  from  Variation  B, 

Calculation  XXII.  D. 

Class  II  (B).     Variation  C  from  the  original  conditions, 

Calculation  XXII.  E. 

On  comparing  them  it  will  be  seen  that  they  all  follow  the 
same  rule,  and  it  will  be  further  noticed  that  the  numerator  of 
the  fraction  in  each  calculation  (log.  4-2191205)  is  the  same, 
and  represents  the  balance  of  original  loan,  £16562'26,  to  be 
provided  by  the  accumulation  of  the  future  annual  increment 
at  the  respective  rates  of  accumulation  in  Calculation  XXII.  C. 
(Variation  A)  at  3  per  cent.,  and  in  Calculations  XXII.  D. 
and  XXII.  E.  (Variation  C)  at  2^  per  cent.  But  in  each 
case  the  above  numerator  (£16562-26,  balance  of  loan)  is 
divided  by  the  amounts  of  £1  per  annum  for  13  years  at 
the  above  respective  rates  per  cent,  of  accumulation,  which, 
as  shown  by  Calculation  (XV)  1,  is  the  usual  method  by  which 
to  obtain  the  sinking  fund  instalment,  as  shown  in  standard 
calculation  form.  No.  3x. 

The  calculation  is  the  same  as  if  it  had  been  assumed  that 
the  present  investments  of  £9932-74  had  been  applied  in  the 
redemption  of  an  equivalent  amount  of  loan  and  an  annual 
sinking  fund  instalment  set  aside  for  the  unexpired  portion  of 
the  repayment  period  of  13  years  to  repay  the  balance  of 
£16562-26  of  original  loan. 

Calculated  in  this  manner,  the  annual  instalment  would  be, 
in  Variation  C,  Calculation  XXII.  C,  the  annual  increment 
of  £1060-474,  which  is  made  up  of:  — 

The  annual  sinking  fund  instalment  of £712-826 

l)lus  the  interest  upon  the  loans  repaid  out  of  the 
sinking  fund  which  (as  pointed  out  in  con- 
sidering the  case  of  local  authorities  in 
Chapter  XIII),  should  be  paid  into  the  fund       £347-648 


£1060-474 


The  same  remarks  apply  equally  to  Variation  C,  as  shown 
by  Calculations  XXII  D.  and  XXII.  E.  On  comparing  these 
calculations,  both  of  which  relate  to  Variation  C,  and  referring 
to  Statement  XXII.  A.,  the  onlv  difference  is  found  in  XXII.  E. 


272    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

(Variation  C,  as  compared  with  tlie  original  conditions)  which 

requires  an  orig^inal  annual  increment  of      £1027'882 

whereas  in  XXII.  D.  it  is  compared  with  Variation 

B.,   which   requires    an    annual    increment   of     £1060'474 


a  difference  of      £32-592 


which  is  the  increased  annual  instalment  required  in  Varia- 
tion A,  as  compared  with  the  original  conditions  in  consequence 
of  the  reduction  in  the  rate  of  accumulation  from  3^  to  3  per 
cent,  in  Variation  A.  From  the  above  data,  as  shown  by 
Calculation  XXII.  C.  and  the  deductive  method  previously 
described  in  Chapter  XIX  relating  to  Variation  A,  it  is  possible 
to  deduce  a  further  method  of  finding  the  future  or  amended 
annual  instalment  consequent  upon  a  variation  in  the  rate  of 
accumulation  accompanied  or  not  by  a  change  in  the  rate  of 
income  to  be  received  upon  the  present  investments  representing 
the  fund.  This  has  been  called  the  annual  increment  (balance 
of  loan)  method,  and  the  rule  may  be  stated  as  follows  :  — 


(1)  From  the  amount  of  loan  repayable  at  the  end 

of  the  original  period  of  repayment  (25  years)     £26495' 00 

(2)  deduct   the   value   of   the   present   investments 

representing  the  fund  at  the  time  the  adjust- 
ment is  required  to  be  made,  namely,  at  the 
end  of  the  12th  year £9932-74 


(3)  and  treat  the  balance  of  loan,  viz.     £16562-26 


as  an  original  amount  to  be  provided  at  the  end  of  the  unexpired 
portion  (13  years)  of  the  original  repayment  period  by  means 
of  an  annual  increment  based  upon  the  future  rate  of  accumula- 
tion. 

(4)  This,  as  shown  by  Calculation  (XIX)  5,  requires 
an  annual  increment  to  be  accumulaled  at 
3  per  cent.,  of     £1060-474 


THE    RATE    OF    ACCUMULATION  273 

(5)  From  this  aunual  sum  deduct  the  future  annual 
income  to  be  received  from  the  present  invest- 
ments at  the  future  rate  per  cent.,  whether 
unaltered,  increased  or  reduced  (in  this  case)       <£o47"648 


(6)  and  the  remainder £712'826 


is  the  future  or  amended  annual  sinking  fund  instalment  to  be 
set  aside  out  of  revenue  or  rate  for  the  unexpired  portion  of  the 
original    repayment   period,    as    ascertained  in    XIX.    A.    and 

XXII.  c. 

In  a  later  chapter  it  will  be  found  that  the  above  rule,  with 
modifications  in  the  wording  only,  may  be  applied  equally  to 
variations  in  the  period  of  repayment  accompanied  or  not  by 
variations  in  the  rates  per  cent,  of  income  and  accumulation. 

This  will  be  shown  in  Chapter  XXIV  dealing  with  a  variation 
in  the  period  of  repayment  only,  and  in  Chapter  XXYI,  dealing 
with  a  concurrent  variation  in  the  period  of  repayment  and  the 
rate  of  accumulation.  For  this  reason  the  summary  of  the 
method  at  the  head  of  this  chapter  has  been  so  worded  that  it 
will  apply  to  the  whole  of  the  problems  above  referred  to. 


274 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


The  Rate  per  cent. 


Calculation  XXII.  C. 


The  Annual  Increment  (ratio)  Method. 
Class  II.  A.  To  find  the  amended  annual  increment  (and 
therefrom  the  additional  annual  instalment)  in  a  sinking 
fund  in  which  the  rate  of  accumulation  is  reduced,  hut  in 
which  the  income  from  the  present  investments,  and  the 
period  of  repayment,  remain  unaltered. 

The  original  conditions  compared  with  Variation  A  by 
the  deductive  method.      Statement  XIX.  A. 

The  rule  relating  to  this  method  is  stated  at  the  head 
of  Chapter  XXIII. 

Required  the  annual  increment  to  he  accumulated  for  a  period 
of  13  years  at  3  per  cent.,  which  is  equivalent  to  an  annual 
increment  of  £1027'882,  to  he  accumulated  for  the  same 
period  at  3|  per  cent. 

Income  from  investments,  3|  per  cent. 


1027-882     -r- 


r  Amount  of  £1  per  annum,  13  years,  3|%   -\ 
Amount  of  £1  per  annum,  13  years,  3%    -'  ~  ' 


or  hy  Table  III,  giving  the  amounts  of  £1  per  annum  : 
1027-882  X  1611303 

,r.r,^o =  1060-474 

15  61(8 

Log.    Present  annual  increment 

add  Log.    Amount  of  £1  per  annum 


1027-882        3-0119434 


Table  III,  13  years,  3i  per  cent.       16-11303         1-2071771 


deduct  Log.  Amount  of  £1 

TqKIo     TTT        1.'^    TTOC,T>C 


jjog.  ^mouni  oi  ^1  per  annum 
Table  III,  13  years,  3 


per  cent. 


16562-26 
15-6178 


4-2191205 
1-1936196 


Log .    Am  ended  annual  increment 

Amended  annual  increment    

To  find  the  amended  annual  instalment: — - 

deduct    the    income    from    investments,    3^ 
per    cent 

Amended  annua]  instalment 

being  Original  annual  instalment  ...  680234 
Additional   annual   instalment     ;')2-592 


3-0255009 
1060-474 


347-648 


712-826 


712-826 


THE    RATE    OF    ACCUMULATION 


"21^ 


The  Rate  per  cent. 


Calculation  XXII.  D. 


The  Annual   Increment  (ratio)  Method. 

Class  IT .  A.  To  find  tlie  amended  annual  increment  (and 
therefrom  the  additional  annual  instalment)  in  a  sinking 
fund  in  which  the  rate  of  accumulation  is  reduced,  but  in 
which  the  income  from  the  present  investments,  and  the 
period  of  repayment,  remain  unaltered. 
Variation  B  compared  with  Variation  C. 

This  calculation  is  exactly  similar  in  principle  to 
XXII.  c. 

The  rule  relating  to  this  method  is  stated  at  the  head 
of  Chapter  XXIII. 

Required  the  annual  increment  to  be  accumulated  for  a  period 
of  13  years  at  2^  per  cent.,  which  is  equivalent  to  an  annual 
increment  of  £*1060'4T4,  to  be  accumulated  for  the  same 
period  at  3  per  cent. 

Income  from  investments,  3  per  cent. 

r Amount  of  £1  per  annum,  13  years,  3%     1 

1060-474    -r ,    „^/ ^ ^5T°r  [=1093-909 

I,  Amount  ot  i/1  per  annum,  13  years,  i\/o   ) 

or  by  Table  III,  giving  the  amounts  of  £1  per  annum  :  — 

1060-474  X  15-6178 

-  =  1093-909 


15-14044 
Log.    Present  annual  increment 
add  Log.    Amount  of  £1  per  annum 
Table  III,  13  years,  3  per  cent. 


deduct  Log.  Amount  of  £1  per  annum 
Table  III,  13  years,  2\  per  cent. 


1060-474 

15-6178 

16562-26 

1514044 


3-0255009 
1-1936196 


4-2191205 
1-1801386 


Log.    Amended  annual  increment  3-0389819 

Amended  annual  increment    1093-909 


To  find  the  amended,  annual  instalment :  — ■ 

deduct    the    income     from     investments,     3 
per   cent 

Amended  annual  instalment 

being  Original  annual  instalment  ...   762-490 
Additional  annual  instalment     33-435 


297-984 


795-925 


795-925 


276    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


The  Rate  per  cent. 


Calculation  XXII.  E. 


The  Annual  Increment  (ratio)  Method. 

Class  11.  B.  To  find  the  amended  annual  increment  (and 
therefrom  the  additional  annual  instalment)  in  a  sinking 
fund  in  which  the  rate  of  accumulation  and  the  income 
from  the  present  investments  are  both  reduced,  but  in 
which  the  period  of  repayment  remains  unaltered. 

The  original  conditions  compared  with  Variation  C. 

The  rule  relating  to  this  method  is  stated  at  the  head 
of  Chapter  XXIII. 
Eequired  the  annual  increment  to  be  accumulated  for  a  period 
of  13  years  at  2\  per  cent.,  which  is  equivalent  to  an  annual 
increment  of  £1027-882,  to  be  accumulated  for  the  same 
period  at  3|  per  cent. 

The  rate  of  income  from  investments  is  reduced  from 
'3|  to  3  per  cent. 

(Amount  of  £1  per  annum,  13  years,  3^% 


1027-882 


=  1093-909 


]  Amount  of  £1  per  annum,  13  years,  2^% 
or  by  Table  III,  giving  the  amounts  of  £1  per  annum  :  — 
1027-882  X  16-11303 


15-14044 


Log.    Present  annual  increment 
add  Log.    Amount  of  £1  per  annum 
Table  III,  13  years,  3|  per  cent. 


=  1093-909 
..       1027-882 
1611303 


30119434 


1-2071771 


deduct  Log.  Amount  of  £1  per  annum 
Table  III,  13  3-ears,  2^  per  cent. 


Log. 


16562-26 
1514044 


4-2191205 
1-1801386 


Amended  annual  increment 

Amended  annual  increment 
To  find  the  amended  annual  instalment  :- 


deduct    the    income 
per   cent 


from     investments,     3 


3-0389819 
1093-909 


297-984 


Amended  annual  instalment 

heiyuj  Original  annual  instalment  ...  680234 
Additional    :nniual  instalment    115-691 


795-925 


795-925 


THE    RATE    OF    ACCUMULATION  277 


CHAPTER  XXIII. 

SINKING     FUND    PEOBLEMS,     EELATING     TO     THE 
EATE  PEE  CENT.  UE  ACCUMULATION  {Contmued). 

Hekivation  of  a  rule  and  formula  relating  to  a  variation 
in  the  rate  per  cent.  of  accumulation  based  upon  the 

FOREGOING  RESULTS  BY  THE  ANNUAL  INCREMENT  (RATIO) 
METHOD. 


The  Annual  Increment  (ratio)  Method. 

The  rule  as  to  a  variation  in  the  rate  of  accumulation  may 
be  stated  as  folloios,  using  the  terms  as  ex  plained  at  the  head 
of  Chapter  XXII.  Statement  XXII.  C. 

EuLE.     7'o  find  the  amended  annual  instalment  to  be  set  aside, 
and  added  to  the  existing  sinking  fund, 

to  be  accumulated  in  future  at  a  rate  per  cent, 
greater  or  less  than  the  rate  at  which  the  present 
annual  instalment  teas  calculated 

{the  future  rate), 
and  to  be  set  aside  during  the  unexpired  portion  of  the 
original  repayment  pcrtod 

iythe  unexpired  period). 

Proceed  as  follows  :  — 

[1)  Ascertain  the  present  annual  increment  of  the  fund,   as 

described  in  Chapter  XXII. 

(2)  Multiply  the  anmial  increment  so  found  by  the  amount 

of  £1  per  annum  at  the  past  rate  for   the  unexpired 
period. 

{3)  Divide  the  above  product  by  the  amount  of  £1  per  annum 
at  the  future  rate  for  the  same  micxpired  period. 

[4)  The  amount  so  found  will  represent  the  future  or 
amended  annual  increment  of  the  fund  under  the  new 
conditions.  The  amended  annual  sinkirig  fund  instal- 
ment may  be  found  by  deducting  therefrom  the  future 
anmial  income  from  the  present  investments  representing 
the  fund. 


278         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 

(Jj  Prepare  a  statement  shoioing  the  jinal  repayment  of  the 
loan  by  the  operation  of  the  sinhiny  fund  under  the 
amended  conditions .  Statement  XIX.  B. 

[6)  Prepare  the  usual  pro  forma  account  previously  recom- 
mended. Pro  forma  Account  3  o.  7. 

This  rule  icill  not  apply  to  cases  in  which  the  rate  of  income 
on  investments  only  is  varied.  Such  problems  may  be  solved 
by  the  simjjle  direct  method,  without  calculation,  described  in 
Chapter  XX,  Statement  C.  It  is  imperative,  in  using  this 
method,  that  the  future  rate  of  accumulation  and,  the  rate  of 
income  upon  the  present  investments  shall  be  uniform  during 
the  whole  of  the  unexpired  portion  of  the  period  of  repayment. 


The  Annual  Inchemext  (eatio)  Method.  Derivation  of  a 
rule  and  formula  relating  to  a  variation  in  the  rate  per  cent,  of 
accumulation.  The  previous  chapters  illustrate  the  various 
methods  of  adjusting  the  annual  sinking  fund  instalment  in 
consequence  of  all  possible  combinations  of  changes  in  the  rates 
per  cent,  of  income  and  accumulation,  with  the  result  that  the 
variations  have  been  divided  into  two  broad  groups  depending 
upon  the  future  rate  of  accumulation,  as  shown  in  Statement 
XXII.  B.  All  variations  relating  to  the  rate  of  income  and 
accumulation  may  be  adjusted  by  making  the  calculation  by 
the  deductive  method  described  in  Chapter  XIX,  but  where  the 
variation  affects  only  the  rate  of  income  upon  investments,  and 
the  rate  of  accumulation  remains  unaltered,  the  deductive 
method  is  superfluous  and  may  be  replaced  by  the  more  simple 
direct  method  without  calculation,  as  described  in  Chapter  XX, 
Statement  XX.  C.  The  variations  affecting  the  rate  of 
accumulation  have  been  divided  into  two  sub-classes  according 
as  the  variation  in  the  rate  of  accumulation  is  accompanied  or 
not  by  a  change  in  the  rate  of  income  upon  the  present  invest- 
ments. 

The  effect  of  a  variation  in  the  rate  of  income  upon  the 
present  investra'ents  has  been  eliminated  by  ascertaining  the 
actual  amount  of  such  income  to  be  yielded  annually  in  future, 
and  treating  the  same  as  an  anunuity  to  be  paid  into  the  fund 
and  accumulated  along  with  the  amended  annual  instalment. 
These  two  annual  sums  have  been  combined  under  the  term 
annual  increment  which  is  acted  upon  by  the  rate  of  accumula- 
tion only,  and  the  enquiry  is  therefore  confined  to  the  rate  of 
accumulation. 


THE    RATE    OF    ACCUMULATION  279 

By  this  method  the  original  annual  instalment,  as  such, 
takes  only  a  minor  place  in  the  calculation  which  is  made  in 
terms  of  the  annual  increment.  Having  found  the  future  or 
amended  annual  increment  required,  under  the  new  conditions, 
to  be  paid  into  the  fund  and  accumulated  for  the  unexpired 
portion  of  the  original  repayment  period,  the  future  annual 
income  from  investments  is  deducted  therefrom  in  order  to 
ascertain  the  future  or  amended  annual  instalment  to  be  set 
aside  out  of  revenue  or  rate.  The  difference  between  this 
amended  instalment  and  the  original  instalment  is  the 
additional  annual  charge  to  revenue  or  rate  due  to  the  variation 
in  the  rates  of  both  income  and  accumulation. 

Having  reduced  all  problems  to  terms  of  the  present  annual 
increment  at  the  date  of  making  the  adjustment,  it  is  found 
that  this  annual  sum  must  be  increased  or  reduced  in  a  definite 
ratio  depending  upon  the  original  and  amended  rates  of 
accumulation.  If  it  be  required  to  ascertain  the  respective 
amounts  of  principal  which  will  provide  a  given  annual  sum 
in  perpetuity  at  two  varying  rates  per  cent.,  they  will  be 
inversely  proportional  to  the  respective  rates.  But  if  it  be 
required  to  find,  as  in  the  problems  now  under  discussion  the 
respective  annuities  which  will  amount  to  a  given  sum  at  the 
end  of  a  given  term  at  varying  rates  per  cent.,  the  element  of 
accumulation  enters  into  the  calculation,  although  the  resulting 
annuities  are  still,  in  a  sense,  in  inverse  ratio  to  the  rates  per 
cent.  Very  little  consideration  will  show  that  the  ratio,  instead 
of  being  expressed  in  terms  of  the  actual  rates  per  cent.,  must 
be  expressed  in  terms  of  the  amounts  of  £1  per  annum  at  the 
respective  rates  per  cent.,  both  for  a  number  of  years  equal  to 
the  unexpired  portion  of  the  period  of  repayment.  This  latter 
provision  is  important;  it  is  not  the  factor  (R)  so  often  used 

(which  is  £1  increased  by  interest  for  one  year)  but       , 

in  which  N  represents  the  number  of  years  in  the  unexpired 
portion  of  the  repayment  period,  and  which  expresses  the 
amount  of  an  annuity  of  £1  in  any  number  of  years,  as  shown 
in  Chapter  VI,  dealing  with  Table  III.  In  the  previous 
discussion  of  the  subject  in  Chapter  XXII  this  method  has 
been  applied  to  three  of  the  examples  previously  considered, 
and  results  have  been  obtained  identical  with  those  found  by 
the  deductive  method.  These  results  are  shown  in  Calculations 
XXII.  C,  D.,  and  E.  On  referring  to  these  calculations  it 
will  be  seen  that  in  each  case  the  actual  working  is  prefaced 
by  a  formula  commencing  with  the  present  annual  increment 


28o    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

at  the  time  the  adjustnieut  is  required  to  be  made,  which  annual 
increment  is  multiplied  by  a  fraction.  In  all  cases  the 
numerator  of  this  fraction  is  the  amount  of  £1  per  annum  at 
the  past  rate  of  accumulation  governing  the  above  annual 
increment  up  to  the  time  of  making  the  adjustment.  The 
denominator  of  the  fraction  is,  in  each  case,  the  amount  of 
£1  per  annum  at  the  future  or  substituted  rate  of  accumulation 
which  will  govern  the  futiire  or  amended  annual  increment 
required. 

The  following  table  will  make  the  matter  clear  and  will  be 
useful  for  future  reference  when  considering  the  question  of  a 
variation  in  the  rate  per  cent,  of  accumulation  accompanied 
by  a  variation  in  the  period  of  repa^anent.  It  shows  the 
respective  variations  in  the  rate  of  accumulation  in  the 
examples  previously  used  to  illustrate  the  derivation  of  a  rule 
and  formula  applying  to  all  such  variations,  namely,  the 
annual  increment  (ratio)  method:  — 

Rate  of  Accumulation  Amount  of  £1  per  annum, 

reduced  : —  for  13  years. 

Calculation.  From  To  Numerator.  Denominator. 

XXII,  C.   oi  per  cent.   3  per  cent.   o^  per  cent.   3  per  cent. 

XXII.  D.     ;}'      „  21     „        ;j       „  2i     „ 

XXII.  E.      :ji       „  21       „  31       „  21       „ 


In  the  whole  of  the  progressive  examples  used  to  illustrate 
the  consideration  of  the  general  question  of  variations  in 
the  rates  per  cent,  of  income  upon  investments  and  of  accumula- 
tion, a  gradual  reduction  in  both  rates  has  been  assumed.  It 
has  been  frequently  pointed  out  that  the  methods  already 
adopted  will  apply  equally  to  an  increase  in  such  rates,  and 
an  inspection  of  Statement  XXII.  B.  will  confirm  this.  It 
will  be  seen  later,  in  Chapter  XXYI,  when  considering  the 
question  of  a  variation  in  the  rate  of  accumulation,  complicated 
by  a  variation  in  the  period  of  repayment,  that  the  sam.e  rule 
holds  good,  seeing  that  the  numerator  of  the  fraction  is  always 
based  upon  the  past  rate  of  accumulation,  and  the  denominator 
upon  the  future  rate. 

A  rule  and  formula  may  now  be  stated,  based  upon  the 
foregoing  considerations  and  upon  Calculations  XXII.  C,  D., 
and  E.,  for  finding  by  direct  calculation  from  the  present 
annual  increment  (not  the  annual  sinking  fund  instalment)  the 
future  or  amended  annual  increment  due  to  a  variation  in  the 
rate  of  accumulation,  whether  accompanied  or  not  by  a  variation 
in  the  rate  of  income  upon  the  investments  representing  the 


THE    RATE    OF    ACCUMULATION 


281 


fund  at  the  time  of  making  the  adjustment.  In  stating  the  rule 
and  formula  relating  to  a  variation  in  the  rate  of  accumulation 
in  this  chapter,  as  well  as  the  rules  relating  to  a  variation  in 
the  period  of  repayment  in  Chapter  XXV,  and  a  concurrent 
variation  in  both  period  and  rate  of  accumulation  in  Chapter 
XXYI,  the  abbreviated  terms  which  are  given  at  the  head  of 
Chapter  XXII  will  be  used,  as  follows  :  — 

The  Past  Rate  denotes  the  rate  of  accumulation  upon  which 
was  based  the  original  annual  instalment  included  in  the 
present  annual  increment. 

The  Future  Rate  denotes  the  rate  of  accumulation  to  be 
used  instead  of  the  past  rate,  to  calculate  the  amended  annual 
increment.  It  will  be  the  same  as  the  past  rate  in  problems 
involving  a  variation  in  the  period  of  repayment  only  without 
any  variation  in  the  rate  of  accumulation. 

The  Unexpired  Period  denotes  the  unexpired  portion  at  the 
time  of  making  the  adjustment  of  the  original  repayment  period 
upon  which  the  present  or  original  annual  instalment  Avas 
based. 

The  Substituted  Period  denotes  the  increased  or  reduced 
number  of  years  over  which  the  future  or  amended  annual 
instalment  shall  be  spread,  and  at  the  end  of  which  the  full 
amount  of  the  loan  will  be  repayable.  It  will  be  the  same  as 
the  unexpired  period  in  problems  involving  a  variation  in  the 
rate  of  accumulation  only,  without  any  variation  in  the  period 
of  repayment. 

The  rule  as  to  a  variation  in  the  rate  of  accumulation  only 
(the  annual  increment  (ratio)  method)  is  stated  in  full  at  the 
head  of  this   chapter. 

The  above  rule  is  sufficiently  explicit,  but  as  it  will,  in 
Chapter  XXVI,  be  combined  with  the  rule  relating  to  a  varia- 
tion in  the  period  of  repayment,  it  is  expressed  as  a  formula 
as  follows  :  — 

Variation  in  the  Rate  of  Accumulation. 

The  Annual  Increment  {ratio)  Method. 

'  Amount  of  £1  per  annum 

at  past  rate 

for  unexpired  period. 


Present 

annual 

increment. 


Amount  of  £1  per  annmn 

at  f^iture  rate 

for  unexpired  period. 


Future 

or 
amended 

annual 
increment. 


282    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

The  amounts  of  £1  per  aunum  iu  the  above  rule  and  formula 
are  at  varying  rates  per  cent,  of  accumulation,  but  are  for  the 
same  number  of  years. 

Calculation  XXII,  C.  will  now  be  expressed  in  terms  of  the 
above  formula,  but  in  this  case  the  problem  will  be  inverted 
to  apply  to  an  increase  in  the  rate  of  accumulation  instead  of  a 
decrease,   as  follows  :  — 

/15-61779' 
1060-474  X  1,,.^^-^.  I   =  1027-882. 


/lO  Di<  ii}\ 

60-474  X  (i^.^TT^.  I   =  1027-^ 
\lo-ii-jOo/ 


In  Chapter  XXVI  this  calculation  will  be  combined  with  the 
similar  calculation  showm  in  Chapter  XXY,  but  relating  to  a 
variation  in  the  period  of  repayment. 

It  Avill  be  noticed  that  the  above  rule,  and  the  formula 
expressing  it,  do  not  contain  any  reference  to  the  future  rate 
of  income  to  be  yielded  by  the  present  investments  representing 
the  fund,  and  that  the  sole  governing  factor  is  the  varying 
rate  of  accumulation.  This  rule  and  formula  will  apply 
equally  to  an  increase  or  decrease  in  the  future  rate  of 
accumulation,  and  it  i.s  important  to  remember  that  an  increase 
in  the  rate  of  accumulation  will  cause  a  reduction  iu  the  annual 
instalment  to  be  charged  to  revenue  or  rate  account  in  future 
years;  an  increase  in  the  repayment  period  will,  on  the  other 
hand,  involve  a  decrease  in  the  future  annual  instalment. 

The  object  of  expressing  the  above  rule  in  formula  form 
will  be  seen  later  in  Chapter  XXV,  when  discussing  the 
adjustment  of  the  annual  instalment  in  consequence  of  a 
variation  in  the  period  of  repayment  only,  and  also  when 
discussing,  in  Chapter  XXVI,  the  adjustment  in  the  annual 
instalment  due  to  a  variation  in  the  period  of  repayment 
accompanied  by  a  variation  in  the  rate  of  accumulation. 

In  Chapter  XXVI  both  the  above  formulae  Avill  be  combined, 
but  in  this  case  Calculation  XXII.  C.  will  be  used  in  an 
inverted  form  in  order  to  obtain  an  example  of  an  increase  in 
the  rate  of  accumulation  from  3  to  '3|  per  cent,  which  will  be 
used  as  the  basis  of  Calculation  XXVI.  C. 

On  comparing  the  above  formula  with  the  formula  in 
Chapter  XXV,  relating  to  a  variation  in  the  period  of  repay- 
ment, it  will  be  noticed  that  the  denominator  in  the  above 
formula  is  the  same  as  the  numerator  in  the  formiila  in 
Chapter  XXV. 


THE    REDEMPTION    PERIOD  283 


CHAPTER  XXIV. 

SINKING    FUND    PROBLEMS,    RELATING    TO    THE 
REDEMPTION  PERIOD. 

A  VARIATION  IN  THE  PERIOD  OF  REPAYMENT  WITH  OR  WITHOUT 
ANY  VARIATION  IN  THE  RATES  PER  CENT.  OF  INCOME  OR 
ACCUMULATION.  SuMMARY  OF  METHODS.  GENERAL  CON- 
SIDERATIONS AS  TO  THE  REDEMPTION  PERIOD.  ThE  DEDUCTIVE 
METHOD.       The    annual    INCREMENT     (rATIO)     METHOD,     AND 

the    annual    increment     (balance    of    loan)    method. 
Statement  showing  the  final  repayment  of  the  loan  by 

THE  operation   OF   THE   AMENDED   ANNUAL   INSTALMENT. 


Summary  of  the  methods  of  adjustment. 

(/)   The  deductive  method,  as  summarised  below  {see  note). 

Statement  XXIV.  A. 

{II)  The  direct  method,  without  calculation,  as  sunnmarised 
at  the  head  of  Chapter  XX,  will  not  apply  to  this  cariation. 

{Ill)  The  annual  increment  [balance  of  loan)  method^  as 
suTnmnarised  at  the  head  of  Chapter  XXll. 

Statement  XXIY .  D. 

{IV)  The  annual  increment  {ratio)  method,  as  summarised 
at  the  head  of  Ch.apter  XXV.  Statement  XXIV.  C. 

Note.  The  terms  used  in  the  following  summary  are  fully 
explained  at  the  head  of  Chapter  XXII.  The  deductive  method 
summarised  below  relates  only  to  a  variation  in  the  period  of 
repoAfment,  and  is  of  limited  application^  in  that  the  rates  of 
accumulation  and  of  income  from  investments  are  both  the 
same  and  remain  unaltered.  The  metliod  described  in  Chapter 
XIX  is  more  generally  applicable,  and  should  be  folloived  in 
all  cases. 

Summary  of  the  deductive  method,  of  ascertaining  the 
amended  annual  sinking  fund  instalment  due  to  a  variation 
in  the  period  of  repaymeiit  only,  without  any  variation  in  the 
rates  per  cent,  of  acciirmilation  or  of  income  from,  the  present 
investments  representing  the  fund,  both  of  which  must  be  the 
same.  Statement  XXIV .  A. 


284    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

(i)  Ascertain  the  vahie  of  the  'present  investments  as 
previously  described. 

[2)  Calculate  the  amount  thereof,  if  accumulated  for  tlie 
substituted  repayment  period  at  the  past  unaltered  rate 
of  accumulation .  Calculation  [XXIV)  1. 

{3)  Calculate  the  ainouiit  of  an  annuity  equal  to  the  present 
or  oriyinal  annual  instalment  for  the  substituted  period 
at  the  past  unaltered  rate  of  accumulation. 

Calculation  [XXI  \')  2. 

(4)  The  amount  found  in  [2)  added  to  the  amount  found  in 

(3)  will  represent  the  amount  of  oriyinal  loan  ivhich  will 
be  provided  thereby  at  the  end  of  the  substituted  period 
of  repay inent. 

(5)  Deduct  the  sum  found  in  (4)  from  the  amount  of  oiiymal 

loan,  and  the  remainder  represents  the  portion  of 
oriyinal  loan  which  will  be  unprovided  for  by  the 
accumtilation  of  the  present  investments  and  the  present 
or  oriyinal  annual  instalment  at  the  past  unaltered  rate 
of  accurtiulation. 

(6)  Calculate  the  additional  annual  sinkiny  fund  instalment 

which  at  the  past  unaltered  rate  of  accmnulation  will 
amount  to  the  balance  of  loan  found  in  (J)  at  the  end  of 
the  substituted  period  of  repayment. 

Calculation  [XXIV)  3. 

(7)  The  additional  annual  instalment  found  in   [6)  added  to 

the  oriyinal  or  present  annual  instalment,  as  in  {3),  yives 
the  future  amended  annual  instalment  to  be  set  aside  and 
added  to  the  fund  duriny  the  substituted  period  of 
repayment. 

{8)  Prepare  a  statement  shoioiny  the  final  repayment  of  the 
loan  by  the  operation  of  the  fund  under  the  amended 
conditions.  Statement  XXJY .  B. 

(9)  Prepare  a  pro  forma  account  showing  the  amount  which 
should  be  in  the  fund  at  the  end  of  each  year  of  the 
substituted  repayment  period. 

Pro  forma  Account,  No.  JO. 

Memo.     The  aborc  method  vill  apply  equally  to  an  increase 
or  reduction  in  the  period  of  repayment. 


THE    REDEMPTION    PERIOD  285 

General    Considerations.       It    very    rarely    happens    that 
there    is    any    alteration    in    the    period    originally    allowed 
for    the     repayment     of     any     individual     loan     of     a     local 
authority.       It    may     be     taken     as     a     general     rule     that 
in  the  special  or  general  Act,  provisional   order,   or  sanction 
of  the  Local  Government  Board,  authorising  the  expenditure 
and    the    consequent    borrowing,    there    is    a    specified    period 
imposed  for  the  final  repayment  of  the  loan  out  of  revenue 
or   rate,    and   this   period   is   strictly   adhered   to.     The   Local 
Government  Board  have  power  under  the  Local  Government 
Act,    1888,    and    the    Public    Health    Acts    Amendment    Act, 
1890,  to  extend  or  vary  the  periods  within  which  loans  may  be 
discharged,  but  this  power  is  limited  to  the  consolidation  of 
debt,  and  the  exercise  of  such  power  is  therefore  confined  to 
the  equation  of  the  repayment  periods  of  the  several  loans  sa 
consolidated.     The  discussion  of  this  part  of  the  subject  will 
be  deferred  to  Chapter  XXXII,   where   it  will  be  fully   con- 
sidered.    It  is  different  with  the   sinking   funds   set  aside  to 
repay  the  loan  debt  of  commercial  or  financial  undertakings. 
In  these  cases  the  conditions  are  much  more  elastic  than  in  the 
case  of  local  authorities,  and  almost  every  kind  of  variation  is 
met  with  in  practice.     These  problems  may  arise  at  the  time  the 
sinking    fund    is    inaugurated    in    order  to    meet    any    special 
obligations  imposed  at  the  time  the  loan  is  arranged,  or  to  meet 
any   future    contingency,    which  it    is    anticipated    may    arise 
during   the   continuation   of  the   fund.     It   may   also   happen 
that  events  occur  after  the  fund  has  been  in  operation  for  some 
years   which   require   that   the   period   of   repayment   shall   be 
increased  or  reduced,  and  any  alteration  in  the  period  may  be, 
and  generally  is,  accompanied  by  a  variation  in  the  rate  per 
cent,  of  accumulation.     Any  variation  in  the  rate  of  interest 
payable   to  the   loan  holders   rarely   affects   the   sinking   fund 
instalment,  and  may  generally  be  ignored,  but  in  all  questions 
of  this  nature  it  is  most  important  to  ascertain  the  whole  of 
the  conditions   in   order   that   the  proper   adjustment   may  be 
made. 

The  Methods  of  Adjustment.  The  dedtictlve  method. 
Although  a  shorter  method  has  been  found  of  making  the 
adjustment  in  the  annual  instalment,  in  the  present  instance 
the  deductive  method  will  again  be  first  used,  afterwards 
making  the  same  adjustment  by  the  methods  described  as  the 
annual  increment  (ratio)  method  and  the  annual  increment 
(balance  of  loan)  method. 


286         REPAYMENT   OF   LOCAL  AND   OTHER  LOANS 

In  tliis  chapter  tlie  variation  will  be  assumed  to  relate  only 
to  the  period  of  repayment  without  any  complication  arising  in 
consequence  of  a  variation  in  the  rate  of  accumulation  or  of 
income  upon  the  present  investments.  In  the  following 
chapter  (XXV)  the  annual  increment  (ratio)  method  will  be 
reduced  to  a  rule  and  formula  relating  to  the  period  of  repay- 
ment only,  in  a  similar  manner  to  that  adopted  in  Chapter 
XXIII,  relating  to  the  rate  of  accumulation.  It  will,  however, 
sometimes  happen  that  an  adjustment  is  required  to  be  made 
owing  to  a  concurrent  variation  in  the  rate  of  income  to  be 
received  from  the  present  investments  and  also  from  the  invest- 
ment of  the  future  accretions  to  the  fund,  and  these  again  may 
be  at  different  rates.  All  questions  arising  out  of  a  variation 
in  the  rate  per  cent,  generally,  have  been  considered  in  previous 
•chapters,  and  the  adjustment  due  to  a  simultaneous  variation 
in  both  period  and  rate  per  cent,  will  be  deferred  to  Chapter 
XXVI. 

The  present  problem  will  be  illustrated  by  the  now  familiar 
example  of  the  sinking  fund  already  discussed,  which  relates 
to  the  repayment  of  a  loan  of  £26,495  at  the  end  of  25  years, 
requiring  an  annual  instalment  of  £680'284,  to  be  set  aside  and 
accumulated  at  3^  per  cent.,  as  found  by  Calculation  (XV)  1. 

Circumstances  have  arisen  which  impose  upon  the  under- 
taking the  necessity  to  accelerate  the  final  redemption  of  the 
loan  indebtedness  by  the  operation  of  the  fund.  It  is  not 
necessary  to  enquire  into  the  special  reason  for  such  acceleration 
because  the  principle  is  the  same  in  any  event.  The  adjustment 
will  again  be  based  upon  the  position  of  the  fund  at  the  end 
of  the  12th  year.  The  undertaking  or  company  was  originally 
required  to  repay  the  loan  of  £26,495  at  the  end  of  the  25th 
year,  namely,  in  13  years  from  the  present  time,  and,  towards 
this,  there  is  in  the  fund  the  proper  calculated  amount,  which 
is  represented  by  investments  valued  at  £9932*74,  as  found  by 
Calculation  (XV)  2,  yielding  an  assured  future  annual  income, 
at  3|  per  cent.,  of  £347"64(S.  The  altered  conditions  demand 
that  the  operation  of  the  fund  shall  be  accelerated  and  that  the 
original  annual  instalment  shall  be  increased  to  such  an  amount 
as  will  repay  the  loan  in  (S  years  from  the  present  time  instead 
of  at  the  end  of  13  years  thereby  reducing  the  original  repay- 
ment period  from  25  to  20  years. 

This  reduction  in  the  period  affects  the  future  accumulation 
of  the  annual  instalment  of  £680"234,  as  originally  calculated, 
and  also  the  future  accumulation  of  the  amount  of  £993274 
now  in  the  fund.     In  order  to  compare  the  resulting  increased 


THE    REDEMPTION    PERIOD  287 

annual  instalment  with  the  original  instalment  it  will  be 
assumed  that  the  original  estimated  rate  of  accumulation, 
namely,  3^  per  cent.,  will  continue  to  be  received  during  the 
remaining  8  years,  both  as  regards  the  income  from  the  present 
investments  and  the  amended  annual  instalment. 

All  the  present  factors  will  be  again  reduced  to  equivalent 
amounts  of  original  loan  which  will  be  provided  at  the  end  of 
the  substituted  period  of  8  years  by  the  accumulation  of  such 
factors  in  order  to  ascertain,  by  the  deductive  method,  as  shown 
in  Statement  XXIV.  A.,  the  portion  of  original  loan  which 
remains  to  be  provided  by  an  additional  annual  instalment. 
If  the  rate  of  accumulation  remains  unaltered  the  reduction  in 
the  period  of  repayment  will  have  the  effect  of  increasing  the 
annual  instalments  as  originally  calculated.  If,  on  the 
contrary,  the  unexpired  period  of  13  years  be  extended  instead 
of  reduced,  there  will  be  an  apparent  surplus  in  the  fund  which 
will  lead  to  a  reduction  in  the  annual  instalment. 

The  additional  annual  instalment  required,  as  shown  in 
Statement  XXIY.  A.,  by  the  deductive  method,  is  £801-862. 
The  balance  of  loan,  £7258-21,  shown  in  Statement  XXIY.  A., 
which  will  be  unprovided  for  owing  to  the  reduction  in  the 
redemption  period  from  13  years  to  8  years  is  made  up  as 
follows :  — 

Present  investments    £9932-74 


Amount  thereof,    accumulated   for 
13  years  at  3^  per  cent. 

Calculation  (XYII)  2     £15534-38 
Amount   thereof,    accumulated   for 
8  years  at  3^  per  cent. 

Calculation  (XXIY)  1     £13079-53 


£2454-85 


Original  annual  instalment :  — 

Amount  of  £680-234  per  annum, 
accumulated  for  13  years  at  3^ 
percent.  Calculation  (XY)  5     £10960-62 

Amount  of  £680-234  per  annum, 
accumulated  for  8  years  at  3| 
per  cent.      Calculation  (XXIY)  2       £6157-26 


£4803-36 


Balance  of  loan  unprovided  for £7258-21 


288    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

It  lias  tlius  been  ascertained  tliat  tlie  ultimate  amount  of 
loan  which  will  be  unprovided  at  the  end  of  the  substituted 
period  in  consequence  of  the  reduction  in  the  original  redemp- 
tion period  is  £7258"21,  and  this  deficiency  has  been  divided 
between  the  accumulations  of  the  present  investments  and  of 
the  original  annual  instalment.  The  portion  of  the  deficiency 
due  to  the  reduced  accumulation  of  the  present  investments  is 
£2454'85,  and  has  been  expressed  in  terms  of  the  capital  value, 
but  it  may  also  be  expressed  in  terms  of  the  annual  income  of 
£347'648  to  arise  from  the  present  investments,  as  follows  :  — 

Amount   of   £347  648   per  annum    in    13    years    at 

31  per  cent.  Calculation  (XXIY)  4     £5601-66 

Amount    of   £347" 648    per  annum    in    8    years    at 

31  per  cent.  Calculation  (XXIV)  5     £3146-81 


£2454-85 


Statement  XXIV.  A.  shows  that  the  reduction  in  the  period 
of  repayment  from  25  years  to  20  years  (but  with  the  same  rate 
of  accumulation)  taking  place  at  the  end  of  the  12th  year, 
results  in  an  increased  annual  burden  of  £801-862  chargeable 
against  the  revenue  of  the  undertaking.  It  only  now  remains 
to  review  the  operation  of  the  fund  under  the  altered  conditions 
in  order  to  ascertain  that  the  amended  annual  instalment  of 
£1482096  so  found  will  carry  out  the  purpose  of  the  fund, 
namely,  to  repay  the  loan  of  £26,495,  but  at  the  end  of  20 
instead  of  25  years.  This  is  shown  in  Statement  XXIV.  B., 
and  by  the  pro  forma  account,  No.  10. 

The  Annual  Increment  (ratio)  Method .  In  previous 
chapters  dealing  with  each  of  the  variations  in  the  rates  per 
cent,  of  income  and  accumulation,  the  additional  annual 
instalment  was  first  ascertained  by  the  deductive  method,  as 
fully  described  in  Chapter  XIX.  This  method  is  based 
essentially  upon  the  ultimate  separate  accumulation  at  the 
future  rate  of  each  of  the  present  factors  of  the  fund,  namely, 
the  annual  instalment  as  originally  calculated,  the  value  of  the 
present  investments,  and  the  future  income  to  arise  therefrom, 
all  of  which  were  reduced  to  equivalent  amounts  of  original 
loan  which  they  will  indiAndually  provide  at  the  end  of  the 
period  of  redemption.  In  Chapter  XXII  iho  whole  of  these 
adjustments  were  again  made  by  direct  calculations  based  upon 


THE    REDEMPTION    PERIOD  289 

the  auuiial  increnieut  of  the  fund  as  defined  in  Chapter  XIY, 
and  it  was  found  that  by  this  means  it  was  possible  to  simplify 
tbe  calculation  and  eliminate  altogether  the  effect  of  any 
variation  in  the  rate  of  income  to  be  received  in  future  upon 
the  present  investments  representing  the  fund.  It  was  found 
that  there  is  an  exact  ratio  existing  between  the  present  and 
future  annual  increments  depending  upon  the  respective 
amounts  of  £1  per  annum;  and  in  Chapter  XXIII,  relating 
solely  to  the  rate  of  accumulation,  this  method  of  calculation 
was  reduced  to  a  rule  and  formula,  called  the  annual  increment 

(ratio)  method. 

For  the  purpose  of  the  following   adjustment  the  present 

annual  increment,  which  is  the  basis  of  the  calculation,  is  made 

up  as  follows  :  — 

Original   annual   instalment      £680"234 

Income  from  present  investments    34T"648 


£1027-882 


The  above  annual  income  from  the  present  investments,  as 
in  all  adjustments  made  by  this  method,  is  tlie  amount  which 
has  been  received  in  the  past,  and  is  not  the  amount  which  will 
be  yielded  thereby  during  the  substituted  period  of  repayment. 

This  is  one  of  the  fundamental  principles  of  the  annual 
increment  (ratio)  method,  as  fully  explained  in  the  opening 
paragraphs  of  Chapter  XXII.  This  method  will  now  be  applied 
to  a  variation  in  the  period  of  repayment,  as  shown  in 
Calculation  XXIY.  C. 

The  Annual  Increment  (balance  of  lo.\n)  Method. 
It  has  been  found  in  Chapter  XXII,  dealing  with  a  variation  in 
the  rate  of  accumulation,  that  the  future  or  amended  annual 
increment,  and  therefrom  the  future  or  amended  annual 
instalment,  may  be  obtained  by  deducting  the  value  of  the 
present  investments  representing  the  fund,  from  the  total 
amount  of  loan  repayable  at  the  end  of  the  prescribed  period, 
and  treating  the  balance  as  an  original  amount  to  be  provided 
by  an  annual  sum  to  be  accumulated  during  the  unexpired 
portion  of  the  original  repayment  period  at  the  future  amended 
rate  of  accuni illation.  The  annual  sum  so  found  is  the 
equivalent  of  the  future  or  amended  annual  increment,  and  the 
future  or  amended  annual  instalment  under  the  new  conditions 


290         REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 

is  found  by  deducting  tlierefrom  the  annual  income  to  be 
received  in  future  upon  tlie  present  investments  representing 
the  fund,  at  any  rate  per  cent,  whether  increased  or  reduced. 
This  is  the  annual  increment  (balance  of  loan)  method,  and 
although  its  derivation  is  not  described  until  Chapter  XXII,  it 
has  been  used  in  previous  chapters.  Statement  XXIV.  D. 
following  gives  details  of  the  present  example  worked  out  by 
this  method. 

The  Redemption  Period.  Statement  XXIV.  A. 

The  Deductive  Method, 

Showing  the  method  of  adjusting  the  annual  instalment 
in  consequence  of  a  variation  in  the  redemption  period 
without  any  variation  in  the  rate  per  cent,  of  accumulation 
or  of  income  from  the  present  investments,  both  of  which 
rates  are  the  same. 

If  these  rates  are  unequal  or  are  varied  proceed  as  in 
Chapter  XIX.  A. 

Conditions  before  adjustment,  at  end  of  12th  year  : 

Amount  of  original  loan,  repayable  in  25  years         £26,495 
Amount  in  the  fund,  at  end  of  12th  year      ...      £9932"74 
Present   annual   income    (previously)    received 

therefrom,   at  3-|  per  cent.,  per  annum     ...       £347"648 
Present  annual  instalment,  to  be  accumulated 

for  13  years,  at  ^  per  cent £680-234 

Present  annual  increment £102T"882 

Variation  from  the  above  conditions  ;— 

The  period  during  which  the  loan  shall   be  redeemed   is 
reduced   from   13  to  8  years. 

The  substituted   period  of  repayment         8  years. 


Present  investments  (at  end  of  12th  year)  £9932"74 


Equivalent 

amount  of 

oriffinal  loan. 


Amount  thereof,   accumulated   for  8  years  at 

3i  per  cent.                      Calculation  (XXIV)  1     £13079-53 
Original  annual  instalment    £680234 


Amount  of  £680-234  per  annum,   for  8  years 

at  3i  per  cent.  Calculation  (XXIV)  2       £6157-26 

Provision  already  made,  will  repay  loan  of £19236-79 


THE    REDEMPTION    PERIOD 


291 


Additional  annual  instalment  required  : — 

]ialauce,  being-  amount  of  original  loan  unpro- 
vided for,  owing  to  the  above  decrease  in  the 
redemption  period  requiring  an  additional 
annual  instalment  to  be  set  aside  and 
accumulated  for  8  years  at  3^  per  cent. 

Additional  annual  instalment 

Calculation  (XXIY)  3  £801-862 


Amount  of  original  loan 

Amended  annual  increment : — 

Annual  income  from  investments. . 
Amended  annual  instalment 


£347-648 
£1482-096 

£1829-744 


£7258-21 


£26495-00 


The  Redemption  Period. 


Statement  XXIV.  B. 


Showing  the  final  repayment  of  the  loan,  by  the  operation  of 
the  sinking  fund  after  making  the  adjustment  in  the 
annual  instalment  consequent  upon  a  reduction  in  the 
period  ol  repayment,  Avithout  any  variation  in  the  rate  per 
cent,  of  accumulation,  or  of  income  from  the  present 
investments. 


Present  investments   (at  end  of  12tli  year)      ...     . 

Amended  annual  increment : — 

Original  annual  instalment £680-234 

Additional  annual  instalment       ...        801*862 


Equivalent 

amount  of 

original  loan. 

£9932-74 


Total  out  of  revenue £1482-096 

Income  from  investments      347*648 

Total £1829-744 


Amount  thereof,   accumulated  for  8  years   at 

3i  per  cent.  Calculation  (XXIV)  6     £16562-26 


Amount  of  original  loan £26495'00 

Amended  annual  instalment        £1482096 


292         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


The  Redemption  Period.  Calculation  XXIV.  C, 

The  Annual  Increment  (ratio)  Method. 

To  find  the  amended  annual  increment  (and  tlierefrom  the 
amended  annual  instalment)  in  a  sinking  fund  in  which 
the  original  period  of  repayment  is  varied,  accompanied  or 
not  by  any  variation  in  the  rates  of  accumulation  or  of 
income  from  the  present  investments. 

The  rule  relating  to  this  method  is  stated  at  the  head 
of  Chapter  XXV. 

Required  the  annual  increment  to  be  accumulated  for  a  period 
of  8  years,  which  is  equivalent  to  an  annual  increment  of 
£1027-882,  to  be  accumulated  for  a  period  of  13  years,  the 
rate  of  accumulation  in  both  cases  being  3^  per  cent. 

1027-882  (^^^o^^^^  o^  ^1  Pe^  annum,  13  years,  3|%|  _.,^pc)-744 

)  Amount  of  £1  per  annum,  8  years,  3^%    ( 
or  by  Table  III,  giving  the  amounts  of  £1  per  annum. 


1027-882x1611303 
9M05I68 


1829-744 


Log.    Present  annual  increment      ...       1027882         30119434 
add  Log.    Amount  of  £1  per  annum 

Table  III,  13  years,  3i  percent.       16-11303         1-2071771 


deduct  Log.  Amount  of  £1  per  annum 
Table  III,  8  years,  3|  per  cent. 


16562-26 
9-05168 


Log.   Amended  annual  increment 

Amended  annual  increment 

To  find  the  a iiiriidcd.  anniuil  insfal »)ent : — - 

deduct    the     inc(mie    from     investments,     3^ 
per  cent.         

Amended  annual  instalment 

being  PreseTit  annmil  instalment   ...   680-2-)4 
Additiomil  annual    instalment  801-862 


4-2191205 
0-9567296 


3-2623909 


1829-744 


347-648 


1482096 


1482096 


THE    REDEMPTION    PERIOD 


293 


The  Redemption  Period, 


Statement  XXIV.  D. 


The  Annual  Increment  (balance  of  loan)  Method. 

To  find  the  amended  annual  sinking  fund  instalment  consequent 
upon  a  variation  in  the  period  of  repayment  with  or  without 
any  variation  in  the  rate  of  income  to  be  received  from 
the  present  investments  or  in  the  rate  of  accumulation. 

For  Eule  see  Chapter  XXII. 

Amount  of  original  loan  (25  years) £26495- 00 

deduct  amount  in  the  fund  at  the  end  of  the 

12th  year £9932-74 


Balance  of  loan 


...  £16562-26 


Amended  annual  increment,  to  be  added  to  the 
fund,  and  accumulated  at  3|  per  cent.,  to 
provide  this  amount  at  the  end  of  8  years. 

Calculation  XXIV.  C.     £1829-744 

deduct  therefrom  income  to  be  received  from 
the  present  investments,  £9932-74,  at  3| 
per  cent.     


£347-648 


Amended    annual    instalment    

being  Original  annual  instalment  ...  £680-234 
Additional  annual  instalment      801-862 


£1482096 


£1482-096 


294         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


Pro  forma  Sinking  Fund  Account,  No.  10. 

A  A'^ariation  in  the  Redemption  Period. 

Loan  of  £26,495,  repayable  at  the  end   of  25  years. 

Showing  the  final  repayment  of  the  loan,  by  the  operation 
of  the  increased  annual  instalment  of  £1482'096. 


Statement  XXIV.  B. 


Rate  of  accumulation,  3^  per  cent. 


Year. 
1 

Amount  in 

the  fund 

at  beginning 

of  year. 

Income 

received  from 

investments 

3i  per  cent. 

Annual 

sinking  fund 

instalment. 

Amount  in 

the  fund 

at  end 

of  year. 

Year 
1 

2 

3 

2 
3 

4 

The  amount  in  the 

fund  at  the 

end  of 

4 

5 

the  12th 

year,  £9932-744,  is  the 

correct 

5 

6 

calculated  amount, 

as  shown  hj  Calcula- 

6 

7 

tion    (XV)  2,    and 

hy    the    pro 

forma 

7 

8 

account, 

No.  1,  Chapter  XV. 

8 

9 

9 

10 

10 

11 

11 

12 

9932-744 

12 

13 

9932-744 

347 

648 

1482-096 

11762-488 

13 

14 

11762-488 

411 

687 

1482-096 

13656-271 

14 

16 

13656-271 

477 

969 

1482-096 

15616-336 

15 

16 

15616-336 

546 

572 

1482-096 

17645-004 

16 

17 

17645004 

617 

575 

1482-096 

19744-675 

17 

18 

19744-675 

691 

063 

1482-096 

21917-834 

18 

19 

21917-834 

767 

124 

1482-096 

24167-054 

19 

20 

24167-054 

845-850 

1482-096 

26495  000 

20 

21 

21 

22 

22 

23 

23 

24 

24 

25 

25 

THE    REDEMPTION    PERIOD  295 


CHAPTER  XXV. 

SINKING  FUND  PROBLEMS,  RELATING  TO  THE 
REDEMPTION  PERIOD  {Continued). 

Derivation  of  a  rule  and  formula  relating  to  a  variation 
in  the  period  of  repayment  based  upon  the  foregoing 
results  by  the  annual  increment  (ratio)  method. 


The  Annual  Increment  (ratio)  Method. 

The  rule  as  to  a  variation  in  the  period  of  repayment,  may 
he  stated  as  follows^  using  the  terms  explained  at  the  head  of 
Chapter  XXII.  Statement  XXIV.  C. 

Rule.     To  find  the  amended  annual  instalment  to  he  set  aside, 
and  added  to  the  existing  sinking  fu7id, 

to  he  accumulated  in  f^iture  at  the  same  rate  per  cent, 
at  tvhich  the  present  annual  instalment  was  calculated 

(the  future  rate), 
and  to  he  set  aside  for  a  reduced  or  increased  number 
of  years  as  com.pared  loith  the  unexpired  portion  of 
the  original  repayment  period 

(the  suhstituted  period). 
Proceed  as  follows  :  — 

(i)  Ascertain  the  present  animal  increment  of  the  fund,  as 
described  in  Chapter  XXII. 

(2)  Multiply  the  an?iual  increment  so  found  by  the  amount 

of  £1  per  annum  at  the  future  rate  for  the  unexpired 
period. 

(3)  Divide    the    above   product   by    the    amount    of   £1    per 

annum,  at  the  future  rate  for  the  suhstituted  period. 

(4)  The    amount    so    found    will    represent    the    future    or 

amended  annual  increment  of  the  fund  binder  the  new 
conditions.  The  amended  annual  sinking  fund  instal- 
ment may  be  found  by  deducting  therefrom  the  future 
annual  income  from  the  present  investments  representing 
the  fund. 


296         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 

(5)  Prepare  a  statement  showing  the  final  repay inent  of  the 

loan   by   the   operation   of   the   sinking   fund  under   the 
a/mended  conditions.  Statement  XXIV .  B. 

(6)  Prepare  the   usual  pro  forma  account  previously  recom- 

mended. Pro  forma  Account,  No.  10,  Chapter  XXIV. 

It  is  imperative,  in  using  this  method,  that  the  future  rate 
of  accumulation  and  the  rate  of  income  from  the  present  invest- 
ments, shall  he  uniform  dur-ing  the  whole  of  the  substituted 
period  of  repayment. 


The  Annual  Increment  (eatio)  Method.  Derivation  of 
a  rule  and  formula,  relating  to  a  variation  in  the  period  of 
repayment.  The  subject  of  enquiry  in  Chapter  XXIII  is  the 
derivation  of  a  rule  and  formula  by  which  to  ascertain  the 
future  or  amended  annual  increment,  and  therefrom  the 
amended  annual  instalment,  due  to  a  variation  in  the  rate  of 
accumulation  only.  The  present  object  is  to  find  a  similar 
rule  and  formula  Avhich  will  apply  to  a  variation  in  the  period 
of  repayment,  and  the  method  to  be  adopted  will  be  the  same  in 
principle.  In  discussing  the  effect  of  a  variation  in  the  rate 
of  accumulation  upon  the  future  or  amended  annual  increment 
in  Chapter  XIX,  Variation  A,  the  amended  annual  increment 
was  ascertained  by  the  somewhat  roundabout,  although  instruc- 
tive, deductive  method  there  described  (Statement  XIX.  A.). 
This  method  was  used  purposely  in  order  to  emphasise  the 
principles  involved  and  to  show  the  effect  of  the  variation  in 
the  rate  of  accumulation  upon  each  of  the  actual  factors  of  the 
fund,  namely,  the  present  investments,  the  annual  income  to 
arise  therefrom,  and  the  original  annual  sinking  fund  instal- 
ment. This  deductive  method  of  enquiry  was  again  adopted  in 
Chapter  XX  (Variation  13,  rate  of  income  upon  investments), 
and  the  amended  annual  increment  was  ascertained  as  shown  in 
Statement  XX.  A.  In  Chapter  XXI,  the  same  method  was 
applied  to  ascertain  the  amended  annual  increment  due  to  a 
dual  variation  in  the  rates  per  cent,  of  accumulation  and  of 
income  upon  investments  (Variation  C),  and  the  result  is  con- 
tained in  Statement  XXI.  A. 

Chapter  XXII  contains  a  tabular  summary  fXXII.  A.)  of 
the  results  obtained  in  all  the  above  investigations  into  the 
effect  of  variations  in  the  rate  per  cent.  This  summar}-  shows 
that  in  each  of  llie  above  eases  the  original  and  amended  annual 
increments  bear  a  certain  definite  ratio  one  to  the  other;  and 


THE    REDEMPTION    PERIOD  397 

fioiii  this  ratio  it  is  possible  to  derive  a  rule  and  formula  by 
which,  to  derive  the  amended  annual  instalment  directly  from 
the  original  annual  instalment. 

In  the  above  examples  the  period  of  repayment  remained 
unaltered,  and  it  has  been  ascertained  that  any  variation  in  the 
rate  per  cent,  of  accumulation  has  the  effect  of  increasing  or 
reducing  the  present  annual  increment  in  proportion  to  the 
ratio  existing  between  the  amounts  of  £1  per  annum  for  the 
unexpired  portion  of  the  original  repayment  period  at  the  past 
and  future  rates  of  accumulation  respectively.  A  similar 
method  will  now  be  applied  to  the  derivation  of  a  rule  and 
formula  by  which  to  find  the  future  or  amended  annual 
increment,  and  therefrom  the  amended  annual  instalment,  due 
to  a  variation  in  the  period  of  repayment,  the  rate  of 
accumulation  remaining  the  same,  and  it  will  be  demonstrated 
by  means  of  the  results  obtained  in  the  example  just  considered 
in  Chapter  XXIV.  In  this  instance  there  is  a  present  annual 
increment,   receivable  for  13  years,  composed  of  :  — 

Original  annual  instalment £680"234 

Income  from  present  investments       347 "648 


£1027-882 


and  this  present  annual  increment,   if  accumulated  at  3^  per 

cent,  for  13  years,  is  sufficient  to  provide  a  definite  amount  of 

loan  at  the  end  of  that  time.     The  above  annual  income  from 

the  present  investments,   as  in  all   adjustments  made  by  this 

method,  is  the  amount  which  has  been  received  in  the  past, 

and  is  not  the  amount  of  income  which  Avill  be  yielded  thereby 

during  the  unexpired  or  substituted  period  of  repayment.     This 

is  one  of  the  fundamental  principles  of  the  annual  increment 

(ratio)  method,  as  fully  explained  in  the  opening  paragraphs  of 

Chapter  XXII.     For  this  purpose  it  is  not  necessary  to  know 

the  actual  amount  of  the  loan,  but  the  above  present  annual 

increment  may  be  treated  as  a  simple  annuity  certain,   for  a 

period  of  13  years,  to  be  acciimulated  at  3^  per  cent.     It  is 

required    to     ascertain    the    equivalent     annuity     or     annual 

increment  accumulating  at  the  same  rate  to  repay  the  same 

loan,  but  at  the  end  of  a  term  of  8  years  instead  of  at  the  end  of 

13  years.     It  has  already  been  ascertained,  in  Chapter  XXIV, 

that  this  equivalent  annual  increment  is  £1829" 744.     In  the 

case  of  the  previous  Calculations  XXII.  C,  D.,   and  E.,  tlie 


29?         REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 

period  of  repaymeut  remaiued  the  same,  but  the  rate  of 
accumulation  varied ;  consequently  the  ratio  was  expressed  in 
terms  of  the  amounts  of  £1  per  annum  at  the  respective  rates 
per  cent.,  but  for  the  same  number  of  years.  In  the  present 
instance  the  rate  of  accumulation  remains  unaltered,  but  the 
period  of  repayment  is  varied.  Consequently  the  ratio  is 
expressed  in  terms  of  the  amounts  of  £1  per  annum  for  the 
respective  unexpired  and  substituted  periods  of  repayment,  but 
at  the  same  rate  per  cent,  of  accumulation. 

In  the  formula  in  Chapter  XXIII  relating  to  a  variation  in 
the  rate  of  accumulation,  the  numerator  is  the  amount  of  £1 
per  annum  for  the  unexpired  period  at  the  past  rate  of 
accumulation,  and  the  denominator  is  the  amount  of  £1  per 
annum  for  the  same  unexpired  period  at  the  future  rate  of 
accumulation,  thus  taking  as  the  basis  of  the  ratio  the  varying 
rates  of  accumulation.  But  as  the  formula  about  to  be 
ascertained  depends  as  to  its  ratio  upon  the  varying  periods  of 
repayment,  and  there  is  not  any  variation  in  the  rate  of 
accumulation,  the  numerator  becomes  the  amount  of  £1  per 
annum  at  the  rate  of  accumulation  common  to  the  two  periods 
for  the  unexpired  period,  and  the  denominator  becomes  the 
amount  of  £1  per  annum  at  the  same  rate  of  accumulation  for 
the  substituted  period.  Substituting  the  above  terms  for  those 
in  the  previous  formula,  the  amended  formula  is  ascertained 
for  dealing  with  problems  involving  variations  in  the  period  of 
repayment  only,  but  not  at  the  same  time  involving  any 
variation  in  the  rate  of  accumulation.  The  rule  and  formula 
as  to  a  variation  in  the  period  of  repayment  only  will  be 
expressed  in  the  same  abbreviated  terms  used  in  Chapter 
XXIII,  dealing  with  a  variation  in  the  rate  of,  accumulation, 
and  these  abbreviated  terms  should  be  carefully  considered. 
They  are  fully  explained  at  the  head  of  Chapter  XXII.  In 
this  case  there  is  not  any  variation  in  the  rate  of  accumulation, 
consequently  the  past  and  future  rates  are  the  same,  and  are,  in 
effect,  the  past  rate.  This  is  important  when  considering  a 
variation  in  the  period  of  repayment  only,  or  a  concurrent 
variation  in  the  rate  of  accumulation  as  well  as  in  the  period  of 
repayment. 

It  is  therefore  necessary  to  use  the  term  "  future  rate  "  in 
the  after  consideration  of  this  and  the  formula  relating  to  the 
dual  variation  in  rate  and  period. 

The  rule  as  to  a  variation  in  the  period  of  repayment  only, 
the  annual  increment  (ratio)  method,  is  stated  in  full  at  the 
head  of  this  chapter. 


THE    REDEMPTION    PERIOD  299 

As  stated  in  Chapter  XXIII,  the  above  rule  is  sufficiently 
explicit,  but  as  it  will  be  necessary  in  the  following  chapter  to 
combine  it  with  the  previous  rule  relating  to  a  variation  in  the 
rate  of  accumulation  it  will  be  expressed  as  a  formula,  as 
follows :  — 

Variation  in  thp:   Period   of   Repayment. 
The  Aniiual  Increment  (ratio)  Method. 


\ 


Present 

annual 

increment 


'Amount  of  £1  per  annum 

at  future  rate 

for  unexpired  period 

Amount  of  £1  per  annum 

at  future  rate 

for  substituted  period 


Future 

or 
amended 

annual 
increment 


The  amounts  of  £1  per  annum  in  the  above  rule  and  formula 
are  at  the  same  rate  per  cent,  of  accumulation,  but  are  for 
varying  numbers  of  years.  In  this  case  the  future  rate  is  the 
same  as  the  past  rate.  Calculation  XXIV.  C.  may  now  be 
stated  in  terms  of  the  above  formula,  as  follows  :  — 

1027-882  X  f^fi^^f )    =1829-744 
V  9-05168  / 

and  in  Chapter  XXVI  this  calculation  will  be  combined  Avith 
the  similar  calculation  in  Chapter  XXIII. 

The  above  rule  and  formula  will  apply  equally  to  an 
increase  or  reduction  in  the  period  of  repayment,  and  it  is 
important  to  remember  that  an  increase  in  the  period  will  have 
the  effect  of  reducing  the  annual  instalment  to  be  charged  to 
revenue  or  rate  account  in  future  years.  When  considering 
the  rate  of  accumulation  in  Chapiter  XXIII  it  was  found  that 
an  increase  in  the  rate  of  accumulation  will  reduce  the  annual 
instalment  in  future  years.  In  the  following  chapter  (XXVI) 
the  above  formula,  relating  to  a  variation  in  the  period  of 
repayment  will  be  combined  with  the  formula  found  in 
Chapter  XXIII,  relating  to  a  variation  in  the  rate  of  accumula- 
tion, for  the  purpo.se  of  deriving  therefrom  a  formiila  which 
may  be  applied  to  a  concurrent  variation  in  the  period  of 
repayment  and  the  rate  of  accumulation. 

It  will  be  noticed  that  the  numerator  in  the  above  formula, 
relating  to  the  period,  is  the  same  as  the  denominator  in  the 
formula  in  Chapter  XXIII,  relating  to  the  rate  per  cent. 


300         REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 


CHAPTER   XXVI. 

SINKING  FUND  PKUBLEMS,  RELATING  TO  THE 
RATE  PER  CENT.  OF  ACCUMULATION  AND  THE 
REDEMPTION  PERIOD  IN  COMBINATION. 

Summary  of  methods.  General  considerations.  The 
methods  of  ascertaining  the  amended  annual  instalment 
due    to    a   variation    in    both    the    above    factors    in 

COMBINATION.  ThE      DEDUCTIVE      METHOD,       THE      ANNUAL 

INCREMENT    (RATIO)    METHOD,     AND    THE    ANNUAL    INCREMENT 
(balance     of     loan)     METHOD.  STATEMENT     SHOWING     THE 

final  repayment  of  the  loan  by  the  operation  of  the 
amended  annual  instalment. 

Derivation  of  a  rule  and  formula  relating  to  a  dual 
variation   of   this   nature   based   upon   the   foregoing 

RESULTS,  BY  THE  ANNUAL  INCREMENT  (rATIO)  METHOD. 


Summary  of  the  methods  of  adjustment. 

(/)  The  deductive  method,  as  summarised  at  the  head  of 
Chapter  XXIV;  ichich  may  he  compared  with  the  method 
summarised  at  the  head  of  Chapter  XIX. 

Statement   XXTI.   A. 

ill)  The  direct  method,  ivithout  calculation,  as  surmnarised 
at  the  head  of  Chapter  XX,  will  not  apply  to  this  variation. 

{Ill)  The  annual  increment  {balance  of  loan)  method,  as 
summarised  at  the  head  of  Chapter  XXII. 

Statement  XXVI .  H . 

{IV)  The  annual  inclement  {ratio)  method,  as  summarised 
below.  Statement  XXVI.  C . 

Note.  The  terms  used  in  the  following  summary  are  fully 
discussed  at  the  head  of  Chapter  XXII .  It  is  imperative,  in 
using  the  above  methods,  that  the  future  rate  of  accumulation 
and  the  rate  of  income  from  the  present  investments  shall  be 
uniform  during  the  whole  of  the  substituted  period  of  repay- 
Tnent. 


THE    RATE    PER    CENT.    AND    PERIOD  301 

The  Annual  Increment  (ratio)  Method. 

The  rule  as  to  a  conmirrent  variation  in  the  rate  of  accumu- 
lation, as  well  as  in  the  'period  of  repay 7ne7it,  inay  he  stated  as 
follows,  using  the  terms  explained  at  the  head  of  Chapter 
XXI L  Statement  XXV I.  C. 

Rule.     To  find  the  amended  annual  instalment  to  he  set  aside, 
and  added  to  the  existing  sinking  fund, 

to  he  accumulated  in  future  at  a  rate  per  cent, 
greater  or  less  than  the  rate  at  which  the  present 
annual  instalTnent  ivas  calculated 

[the  future  rate), 
and  to  he  set  aside  for  a  reduced  or  increased  number 
of  years,  as  cornpured  ivith  the  unexpired  portion  of 
the  original  repayment  period 

{the  s^ihstituted\  period). 
Proceed  as  follows  :  — 

{!)  Ascertain  the  present  annual  increment  of  the  fund,  as 
described  in  Chapter  XXII. 

(2)  Multiply  the  annual  incrcTnent  so  found  by  the  amount 

of  £1  per  anmim  at  the  past  rate  for  the  unexpired 
period. 

(3)  Divide    the    above    product    by    the    amount    of    £1    per 

annum,  at  the  future  rate  for  the  substituted  period. 

(4)  The    amount    so    found    loHl    represent    the    future    or 

amended  annual  increment  of  the  fund  under  the  new 
conditions .  The  amended  annual  sinlcing  fund  instal- 
ment may  be  found  by  deddicting  therefrom  the  futtire 
annual  income  from,  the  present  investments  representing 
the  fund. 

[6)  Prepare  a  statement  showing  the  flrial  repayment  of  the 
loan  by  the  operation  of  the  fund  under  the  amended 
conditions .  Statement  XXVI .  B. 

(6)  Prepare  the  usual  pro  forma  account  previously  recom- 
mended. Pro  forma  Account,  No.  11. 


General  Considerations.  The  predommant  factor  in  all 
problems  of  tliis  nature  is  the  variation  in  the  period  of 
repayment  because  its  effect  upon  the  amended  annual  instal- 
ment is  far  greater  than  that  due  to  the  variation  in  the  rate 


302  REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 

of  accumulation  which  will  generally  lie  within  very  narrow 
limits.  For  the  reasons  given  in  Chapter  XXIV,  a  variation 
of  this  two-fold  nature  will  rarely  arise  in  connection  with 
any  individual  loan  of  a  local  authority,  and  if  such  a  problem 
arises  in  connection  with  the  consolidation  of  several  such  loans 
it  will  be  complicated  by  other  factors  which  will  render 
necessary  a  different  mode  of  treatment,  as  will  be  explained 
in  Chapter  XXXII,  dealing  generally  with  the  equation  of  the 
period  of  repayment. 

The  principal  application  of  the  methods  to  be  discussed  in 
this  chapter  will  relate  to  the  sinking  funds  of  commercial  and 
financial  undertakings,  and  all  the  general  considerations  as  to 
a  variation  in  the  period  of  repayment  only,  stated  in  Chapter 
XXIV,  will  apply  to  this  example  without  further  reference 
or  amplification. 

In  dealing  with  problems  which  may  arise  in  connection 
with  the  sinking  funds  of  local  authorities  and  commercial  and 
financial  undertakings,  the  following  important  factors  have 
already  been  discussed,  namely  :  — 

1.  The  amount  in  the  fund. 

2.  The  rate  per  cent. — 

(a)  of  income  upon  the  present  investments. 
(6)  of  future  accumulation. 

3.  The  period  of  repayment. 

In  discussing  the  problems  relating  solely  to  a  variation  in 
the  rate  per  cent,  or  the  period  of  repayment',  in  each  case  there 
has  been  combined  in  one  factor,  "  the  annual  increment," 
(1)  the  original  or  amended  annual  instalment,  and  (2)  the  past 
or  future  income  arising  from  the  present  investments  represent- 
ing the  fund  at  the  time  the  variation  occurs  in  the  rate  or 
period. 

This  annual  increment  is  fully  discussed  and  described  in 
Chapters  XIV  and  XXII.  The  majority  of  the  examples  used 
to  illustrate  the  above  problems  relate  to  a  sinking  fund  to 
repay  a  loan  of  £26,495  at  the  end  of  25  years,  and  it  has  been 
assumed  that  ihe  variation,  and  the  consequent  necessity  for 
adjustment,  occurs  at  the  end  of  the  12tli  year  in  each  case. 
As  regards  a  variation  in  the  rate  per  cent.,  it  has  been  proved 
that  the  problem  may  be  confined,  so  far  as  the  actuarial 
calculation  is  concerned,  to  the  rate  of  accumulation.  It  has 
also   been   ascertained   that   there    is    a    simple   ratio    existing 


THE    RATE    PER    CENT.    AND    PERIOD  303 

between  the  original  and  amended  annual  increments  due  to  a 
variation  in  both  the  rate  and  the  period,  and  that  this  ratio 
is  based,  not  upon  the  respective  rates  per  cent,  of  accumulation 
or  upon  the  number  of  years  in  the  period  of  repayment,  but 
upon  the  respective  amounts  of  £1  per  annum  as  follows  :  — 

1.  In  the  case  of  a  variation  in  the  rate  of  accumulation, 
upon  the  amounts  of  £1  per  annum  for  the  same  period  of 
repayment,  but  at  the  respective  rates  per  cent,  of  accumulation. 

Chapter  XXIII. 

2.  In  the  case  of  a  variation  in  the  period  of  repayment, 
upon  the  amounts  of  £1  per  annum  at  the  same  rate  per  cent, 
of  accumulation,  but  for  the  respective  periods  of  repayment. 

Chapter  XXV. 

In  the  case  of  variations  in  the  rate  per  cent,  the  necessary 
adjustment  has  been  made,  in  the  first  instance,  by  the  deductive 
method,  fully  described  in  Chapter  XIX,  based  upon  the  whole 
of  the  factors  governing  the  fund,  after  which  the  result  so 
obtained  has  been  verified  by  the  annual  increment  (ratio) 
method  based  upon  the  annual  increment,  as  described  in 
Chapter  XXII.  These  results  have  been  utilised  to  deduce  a 
rule,  and  a  formula  expressing  the  rule,  which  is  fully  described 
in  Chapter  XXIII. 

The  enquiry  was  then  extended  in  a  similar  manner  to  an 
adjustment  rendered  necessary  by  a  variation  in  the  period  of 
repayment  which  was  considered  in  Chapters  XXIV  and  XXV, 
and  a  similar  rule  and  formula  was  deduced.  In  each  case  it 
was  found  that  the  methods  applied  equally  to  an  increase  or  a 
reduction  in  the  rate  of  accumulation  or  period  of  repayment. 

The  adjustment  consequent  upon  a  dual  variation  in  the  rate 
of  accumulation,  as  well  as  in  the  period  of  repayment,  will  be 
fully  considered  in  the  present  chapter,  using  the  whole  of  the 
methods  already  described,  after  which  a  rule  and  formula 
relating  to  the  adjustment  will  be  deduced  from  the  results  so 
obtained. 

The  Deductive  Method.  The  present  enquiry  will  also 
be  illustrated  by  a  sinking  fund  to  repay  a  loan  of  £26,495  at 
the  end  of  25  years,  but  with  a  rate  of  accumulation  of  3  per 
cent.,  requiring  an  annual  instalment  of  £712''826  to  be  set 
aside  for  the  remaining  13  years.  This  has  been  ascertained 
in  Chapter  XIX,  Statement  XIX.  A. 

The  necessity  to  make  the  adjustment  arises  at  the  end  of 
the  12th  year,  at  which  time  the  amount  in  the  fund  is 
£993274,  which  is  represented  by  investments  valued  at  that 


304         REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 

amount,  bringing  in  an  annual  income  at  3^  per  cent,  per 
annum  of  £347'648,  and  it  will  be  assumed  tbat  this  income  is 
assured  for  the  whole  of  the  unexpired  portion  of  the  original 
repayment  period.  At  the  end  of  the  12th  year,  this  period  is 
for  some  reason  reduced  from  13  years  to  8  years,  and  the  rate 
of  accumulation  is  increased  from  3  to  3^  per  cent.,  as  in  the 
original  conditions  in  Chapter  XY. 

The  effect  will  be  that  the  annual  instalment  of  £712-826 
will  be  increased  in  consequence  of  the  reduction  of  the  period 
of  repayment,  but  it  will  not  be  increased  to  such  an  amount  as 
it  would  have  been  if  the  rate  of  accumulation  had  remained  at 
3  per  cent,,  and  had  not  been  increased  to  3|  per  cent.  This 
will  be  shown  later  in  this  chapter  in  detail,  where  the  amended 
annual  instalment  will  be  divided  between  these  factors.  As  in 
previous  examples  by  the  deductive  method,  all  the  present 
factors  of  the  fund  will  be  reduced  to  equivalent  amounts  of 
orit^inal  loan  which  they  will  each  provide  by  accumulation  at 
the  future  rate  of  3^  per  cent.,  at  the  end  of  the  substituted 
period  of  8  years,  in  order  to  ascertain,  by  deduction,  the 
portion  of  original  loan  which  will  remain  to  be  provided  by  the 
future  accumulation  of  an  additional  annual  instalment  to  be 
charged  to  revenue  or  rate  or  deducted  from  profits,  and  a  final 
calculation  will  be  made  to  ascertain  such  additional  instalment. 

This  is  fully  shown  in  Statement  XXYI.  A.,  which  is  similar 
in  principle  to  previous  statements  illustrating  the  deductive 
method.  This  statement  shows  that  the  reduction  in  the  original 
period  of  repayment  from  25  to  20  years  (but  with  an  increase 
in  the  rate  of  accumulation)  taking  place  at  the  end  of  the 
12th  year  results  in  an  increased  annual  burden  of  £769"'270 
chargeable  against  the  rate  account  or  the  revenue  account  of 
the  undertaking. 

It  is  now  possible  to  review  the  operation  of  the  sinking  fund 
under  the  altered  conditions  in  order  to  ascertain  that  the 
amended  annual  instalment  so  found  will  carry  out  the  purpose 
of  the  fund,  namely,  to  repay  the  loan  of  £26,495,  at  the  end 
of  20  instead  of  25  years.  This  is  shown  in  Statement  XXYI.  B., 
which  is  exactly  similar  in  principle  to  the  previous  statements 
prepared  to  illustrate  the  accuracy  of  the  amended  annual 
instalments  found  by  the  deductive  and  other  methods. 

The  Annual  Increment  (ratio)  Mettiod.  (Tfule  and 
Formula.)  In  Chapter  XXII  (a  variaiiou  in  the  rate  of 
accumulation)  as  well  as  in  Chapter  XXIY  (a  variation  in  the 
pciiod  of  repaymeni)  the  actual  adjustment  has  been  made  bv 


THE    RATE    PER    CENT.    AND    PERIOD  305 

the  annual  increment  (ratio)  method  there  described,  and  from 
the  results  so  obtained  the  formula  relating  to  the  method  has 
been  deduced.  In  both  these  variations  the  ratio  is  a  simple 
one,  depending  upon  the  respective  amounts  of  £1  per  annum  at 
the  varying  rates  per  cent,  in  one  case  and  for  the  varying 
periods  in  the  other.  As  the  rules  and  formulse  relating  to  the 
above  variations  have  been  already  ascertained,  it  is  only 
necessary  in  the  present  instance  to  revert  to  those  formulae  in 
order  to  deduce  therefrom  a  modified  formula  relating  to  a 
combination  of  the  above  causes  of  adjustment,  and  afterwards 
to  make  the  calculation  in  the  manner  shown  in  Chapters  XXII. 
and  XXIV.  It  would  appear  from  the  above  theoretical  con- 
siderations that  the  two  formulae  may  be  combined  in  order  to 
deduce  therefrom  a  simple  formula  which  will  apply  to  all 
problems  involving  a  dual  variation  in  the  rate  of  accumulation 
and  the  period  of  repayment.  It  is  therefore  necessary  to 
combine  the  formula  relating  to  a  variation  in  the  rate  of 
accumulation  given  in  Chapter  XXIII  with  that  relating  to  a 
variation  in  the  period  of  repayment  in  Chapter  XXV.  The 
factors  required  are  :  — 

1.  The  present  annual  increment. 

2.  The  past  and  future  rates  of  accumulation. 

3.  The  unexpired  and  substituted  periods  of  repayment. 

For  the  purpose  of  the  following  adjustment  the  present 
annual  increment,  which  is  the  basis  of  the  calculation,  is  made 
up  as  follows  :  — 

Original  annual  instalment  (Statement  XIX.  A.)  ...  £T12"826 
Income  from  present  investments       347'648 


£1060-474 


The  above  annual  income  from  the  present  investments,  as  in 
all  adjustments  made  by  this  method,  is  the  amount  which  has 
been  received  in  the  past,  and  is  not  the  amount  of  income 
which  will  be  yielded  thereby  during  the  unexpired  or 
substituted  period  of  repayment.  This  is  one  of  the  funda- 
mental principles  of  the  annual  increment  (ratio)  method,  as 
fully  explained  in  the  opening  paragraphs  of  Chapter  XXII. 

The  method  of  making  the  adjustment  by  this  method  is 
shown  in  Calculation  XXVI.  C,  at  the  end  of  this  chapter. 

In  each  of  the  examples  discussed  in  Chapters  XXII  and 
XXIV  the  original   annual   increment  was  multiplied  by  the 


3o6    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

fractional  ratio  of  £1  per  annum.  It  is  therefore  obvious  tliat 
a  combination  of  the  above  formulae  to  relate  to  the  dual 
variation  under  discussion  must  be  made  by  multiplying  the 
present  annual  increment  by  each  fractional  ratio  in  succession. 
As  already  pointed  out,  the  numerator  in  the  fractional  ratio 
relating  to  the  period  of  repayment  is  the  same  as  the 
denominator  in  the  fractional  ratio  relating  to  the  rate  of 
accumulation,  which  will  cancel  out  when  the  respective 
formulae  are  multiplied  together ;  therefore  the  product  of  these 
fractional  ratios  Avill  consist  of  the  numerator  of  the  ratio 
relating  to  the  rate  of  accumulation  and  the  denominator  of  the 
ratio  relating  to  the  period  of  repayment  as  follows  :  — 


Variation  in  the  Rate  of  Accumulation  and  the 
Period  of  Repayment. 

The  Annual  Increment  (^ratio)  Method. 


Variation  in  Bate. 
Chapter  XXJII. 


Variation  in  Period^ 
Chapter  XXV . 


\ 


Present 

annual 

increment 


( Amount  of  £1  per  ann. 
at  past  rate, 
for  unexpired  period 


Amount  of  £1  per  ann. 
at  future  rate, 
for  unexpired  period 


Amount  of  £1  per  ann. 
at  future  rate, 
for  uncrpired  period 


Amount  of  £1  per  ann. 

at  future  rate, 
for  substituted  period 


Future 

or 
amended 
annual 
^increment 


Note.  The  factors  in  the  above  formulae  which  are  printed 
in  italics  are  common  to  both  and  will  cancel  out  in  the  multi- 
plication. Calculation  XXA^I.  C.  may  now  be  stated  in  terms 
of  the  above  formula  in  a  similar  manner  to  that  adopted  in 
Chapters  XXIII  and  XXV:  — 


1000-474  X 


L5-61779( 
ieill808( 


Hill 


9-05168 


308)  _ 

4~ 


=  1829-744 


The  result  is  the  foHowing  simplified  formula  relating  to  a 
concurrent  variation  in  tlie  rale  of  accuniulaiion  and  the  period 
of  repayment :  — 


THE  RATE  PER  CENT.  AND  PERIOD 


307 


Variatiox  in  the  Eate  of  Accumulation  and  the  Period 
OF  Repayment. 

The  Annual  Increment  (ratio)  Mctliod. 
[Amount  of  £1  per  annum 


Present 

annual 

increment 


at  past  rate, 
for  unexpired  period 


Future 

or 
amended 

annual 
increment 


Amount  of  £1  per  annum 

at  future  rate, 
for  substituted  period 

The  amounts  of  £1  per  annum  in  tlie  above  formula  are  at 
varying  rates  per  cent.,  and  are  also  for  different  numbers  of 
years. 

Calculation  XXYI.  C.  will  now  be  expressed  in  terms  of  tbe 
above  formula  as  follows  :  — 


1060-474  X 


(l5-6l779| 
I   9-05168  j 


1829-744 


It  is  now  possible  to  state  a  rule  based  upon  tbe  foregoing 
formula,  using  tbe  abbreviated  terms  set  out  in  full  at  tbe 
head  of  Chapter  XXII,  and  explained  in  Chapter  XXIII, 
dealing  with  the  rule  relating  to  a  variation  in  the  rate  of 
accumulation.  The  same  terms  are  used  in  Chapter  XXY, 
in  the  rule  relating  to  a  variation  in  the  period  of  repayment. 
The  rule  relating  to  the  variation  under  review  is  stated  in  full 
at  the  head  of  this  chapter. 

Proof  of  the  aeove  Method.  The  foregoing  results  which 
have  been  obtained  by  taking  both  variations  into  account  will 
now  be  proved,  and  the  effect  of  each  variation  will  be  shown 
separately,  beginning  with  the.  variation  in  the  period  of 
repayment.  In  Chapter  XXIY,  an  adjustment  Avas  made  in 
the  annual  instalment  consequent  upon  a  reduction  in  the 
period  of  repayment  from  13  to  8  years,  but  without  any 
variation  in  the  rate  of  accumulation.  This  reduction  in  the 
period  involved  an  ultimate  deficiency  of  loan  of  £7258-21, 
requiring  an  additional  annual  instalment  of  £801-862  to  be  set 
aside  for  the  substituted  period  of  8  years,  as  shown  in  Statement 
XXIV.  A.  The  accuracv  of  the  calculation  was  proved  by 
dividing  the  deficiency  in  the  amount  of  loan,  £7258-21, 
between  the  reduced  accumulation  of  the  present  investments. 
£2454-85,   and  of  the   original   annual   instalment,   £4803-o6. 


3o8    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Tile  future  deficiency  in  tlie  accumulation  of  the  present  invest- 
ments, £2454*85,  was  also  reduced  to  terms  of  the  annual 
income  to  arise  therefrom. 

Although  the  same  method  of  proof  may  be  applied  to  the 
present  example  the  problem  will  be  reduced  to  terms  of  the 
present  annual  increment  of  £1060474,  and  by  deducting 
therefrom  the  income  from  investments,  £347'648,  included 
therein,  it  will  be  possible  at  the  same  time  to  express  in  figures 
the  effect  upon  the  additional  annual  instalment  of  the 
reduction  in  the  period  of  repayment,  as  distinguished  from  the 
effect  of  the  increase  in  the  rate  of  accumulation. 

The  calculation  will  be  made  by  the  annual  increment 
(ratio)  method,  which  is  the  most  convenient  for  the  purpose. 
The  problem  Avill  be  divided  into  two  parts  in  order  to  ascertain 
in  the  first  place  the  amended  annual  increment  due  to  the 
reduction  in  the  period  of  repayment  only,  on  the  assumption 
that  the  rate  of  accumulation  remained  the  same,  namely, 
3  per  cent.  This  amended  annual  increment,  as  shown  by  the 
following  Statement  XXYI.  D.,  is  £1862'532,  requiring  an 
additional  annual  instalment  of  £802"058. 

Although  it  will  be  necessary  to  consider  the  above 
additional  annual  instalment  of  £802'058  later  in  this  chapter, 
the  proof  will  be  continued  by  taking  up  the  above  amended 
annual  increment  of  £1862'532  in  order  to  ascertain  the 
reduction  therein  due  to  the  increase  in  the  rate  of  accumulation 
from  3  per  cent,  to  3^  per  cent.  The  calculation  cannot  be 
made  in  terms  of  the  above  additional  annual  instalment  of 
£802-058  for  the  reasons  given  in  Chapter  XXII,  Calculation 
XXII.  E.,  because  the  benefit  of  the  accumulation  of  the 
income  from  the  present  investments  at  the  increased  rate  of 
accumulation  would  be  lost.  The  calculation  might  be  made 
in  terms  of  each  of  the  above  factors,  namely  the  annual 
instalment  and  the  income  from  investments  composing  the 
annual  increment  of  £1862'532,  but  this  would  involve  only 
increased  labour  without  any  corresponding  advantage,  seeing 
that  the  accuracy  of  the  calculation  may  be  proved  by 
comparing  the  additional  annual  instalment  to  \v?  obtained 
with  that  found  by  the  deductive  method,  and  also  by 
comparing  the  amended  annual  increment  with  that  foimd 
previously  by  the  annual  increment  (ratio)  method,  Calculation 
XXVI.  C.  This  method  of  proof  shows  the  advantage  of  the 
annual  increment  as  a  factor  even  in  cases  where  there  is  not 
any  variation  in  the  rate  of  income  from  the  present  invest- 


THE    RATE    PER    CENT.    AND    PERIOD  309 

ments.     This  is  sliowu  in  Statement  XXYI.  E.  at  the  end  of 
this  chapter. 

The  results  obtained  by  the  foregoing  calculations  may  now 
be  summarised  in  order  to  prove  the  accuracy  of  the  previous 
methods  of  adjustment.  The  additional  annual  instalment  of 
£769270  found  in  Statement  XXVI.  E.  agrees  with  that  found 
by  the  deductive  method  (Statement  XXYI.  A.),  and  the 
amended  annual  increment  of  £1829-744  in  Statement  XXVI.  E. 
agrees  with  that  found  by  the  annual  increment  (ratio)  method 
(Statement  XXVI.  C).  Attention  may  now  be  drawn  to  the 
additional  annual  instalment  so  found  in  order  to  ascertain  the 
relative  effects  thereon  of  the  variation  in  each  factor  of  period 
and  rate  per  cent. 

The  original  annual  instalment  accumulated  at 
3  per  cent,  before  these  variations  occurred,  as 
shown  by  Statement  XIX.  A.,  was £712'826 

and  the  result  of  the  reduction  in  the  period  of 
repayment,  taken  by  itself,  is  to  increase  the 
original  annual  instalment  by  an  additional 
annual  amount,  as  shown  by  Statement  XX\  I. 
D.  of    £802-058 

but  the  effect  of  the  increase  in  the  rate  of  accumula- 
tion is  to  reduce  this  annual  amount  bv £32'788 


leaving  a  net  increase  in  the  original  annual 
instalment,  as  shown  by  Statement  XXVI.  E. 
of      £769-270 


Proof  of  Method  (continued).  The  present  example  has 
already  been  compared  with  that  used  in  Chapter  XXIV,  to 
illustrate  the  effect  of  a  variation  in  the  period  of  repayment 
only.  It  was  found  by  Calculation  XXIV.  A.  that  the 
additional  annual  instalment  in  that  case  was  £801-862.  In 
the  present  case,  which  is  complicated  by  a  variation  in  the  rate 
of  accumulation,  an  additional  annual  instalment  of  £802-058 
is  required.  As  in  both  cases  the  reduction  in  the  period  of 
repayment  is  the  same,  namely,  from  13  years  to  8  years,  the 
two  examples  may  be  connected,  bearing  in  mind,  however, 
that  the  rate  of  accumulation  in  the  present  instance  is  3  per 
cent.,  and  that  therefore  the  annual  instalment  at  the  time  of 
making  the  adjustment  is  £721-826  instead  of  £680-234,  and 


3IO    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

a  rate  of  accumulation  of  3^  per  cent.,  as  in  the  previous 
example.     (See  Statement  XIX.  A.) 

In  that  case  it  was  found  by  the  deductive  method  (State- 
ment XXIV.  A)  that  the  ultimate  dehciency  in  the  amount 
of  loan  to  be  provided,  in  consequence  of  the  reduction  in  the 
period  of  repayment,  was  i/7258"21,  Avhich  requires,  as  shown  by 
Calculation  (XXIV)  3,  an  additional  annual  instalment  of 
<£801'8G2  to  be  accumulated  at  o|  per  cent,  for  8  years. 

It  is  therefore  necessary  to  ascertain  the  equivalent  annual 
instalment  to  jjrovide  the  same  amount  of  loan,  i;;7258'21,  at 
the  end  of  8  years,  but  to  be  accumulated  at  3  per  cent,  instead 
of  3;!  per  cent.  This  is  shown  to  be  £816"232  in  Statement 
XXVI.  F. 

Statement  XXVI.  G.  shows  that  if  the  additional  annual 
instalment  of  £816'232,  as  found  by  Calculation  XXVI.  F.  be 
adopted,  there  will,  at  the  end  of  the  substituted  period  of 
8  years,  be  in  the  fund  an  amount  of  i>12G'04  in  excess  of  the 
amount  actually  required  to  repay  the  loan ;  and  therefore  that 
the  additional  annual  instalment  of  i>816"232  must  be  reduced 
by  an  annual  sum  which,  if  accumulated  at  3  per  cent.,  will 
amount  to  £126'04:  at  the  end  of  8  years,  which  is  the  annual 
sinking  fund  instalment  which  will  provide,  or  the  annuity 
which  will  amount  to  that  sum,  under  the  above  conditions. 
By  Calculation  (XXVIj  5  the  annual  sum  is  found  to  be 
£14174. 

The  correct  additional  annual  instalment  required  for  the 
purpose  of  showing  the  separate  eifect  of  the  variation  in  the 
period  therefore  is : — 

The  above  calculated  instalment  of £816232 

reduced  by  the  above  annuity  of     '      £14'174 


leaving  the  actual  additional  annual  instalment  of     £802'058 


which  agrees  Avitli  the  amount  found  by  Statement  XXVI,  D. 
by  the  annual  increment  (ratio)  method. 

The  Annual  Ixchkment  (balance  of  loan)   Method.     In 
previous  chapters  attention  has  been  directed  to  the  principles 

underlying  this  meiliod.  It  I'esembles  very  closely  the  practice 
adopted  by  such  local  authorities  as  are  able  to  apply  the  whole 
of  the  annual  instalments  towards  the  immediate  actual 
redemption  of  debt.     In  such  cases  the  interest  upon  the  debt 


THE    RATE    PER    CENT.    AND    PERIOD  311 

so  redeemed,  and  the  future  annual  instalments,  constitute  the 
annual  increment  of  this  method  provided  the  rate  of  interest 
upon  such  redeemed  debt  is  the  same  as  the  rate  of  accumula- 
tion. In  case  there  is  any  variation  in  these  two  rates  per  cent, 
the  annual  difference  may  be  transferred  as  and  when  it  arises 
to  the  debit  or  credit  of  the  revenue  or  rate  account.  It  is  an 
essential  principle  of  this  method  that  the  resulting  annual 
instalment,  the  future  rate  of  income  from  the  present  invest- 
ments, and  the  rate  of  accumulation  shall  continue  without 
variation  during  the  whole  of  the  unexpired  portion  of  the 
repayment  period.  Any  departure  from  uniformity  in  these 
respects  has  already  been  pointed  out.  Chapter  XYI,  dealing 
with  the  adjustment  of  a  deficiency  in  the  fund  contains  fvill 
particulars  of  the  method  of  finding  the  additional  annual 
instalment  to  be  spread  over  a  portion  only  of  the  unexpired 
repayment  period,  and  Chapter  XXVII  explains  the  method 
of  correcting  the  annual  instalment  in  consequence  of  a 
variation  in  the  future  rate  of  income  to  be  received  from  the 
present  investments,  which  is  expected  to  occur  at  a  future  date 
during  such  unexpired  period,  and  a  similar  method  of  adjust- 
ment will  apply  to  a  variation  in  the  rate  of  accumulation 
occurring  at  such  a  future  date. 

In  dealing  with  such  a  future  variation  in  the  rate  per 
cent,  of  income  or  of  accumulation  in  Chapter  XXVII,  a  dis- 
tinction has  been  drawn  between  a  reduction  which,  although 
anticipated,  is  uncertain  both  as  to  rate  and  time,  and  one  in 
which  both  factors  are  definite,  as  was  the  case  with  the  reduc- 
tion in  the  dividend  on  consols  under  Mr.  Goschen's  Finance 
Act,  1888.  In  Chapter  XXVII  attention  is  also  directed  to 
the  difference  between  the  arithmetical  and  true  mathematical 
methods  of  arriving  at  the  equated  rate  per  cent.,  and  it  is  there 
pointed  out  that  the  same  difference  in  such  methods  occurs  in 
the  equation  of  the  period  of  repayment,  as  will  be  fully 
described  in  Chapter  XXXII.  Any  variations  in  the  above 
factors  of  rate  per  cent,  or  period  of  repayment  anticipated  to 
arise  during  the  unexpired  portion  of  the  repayment  period, 
whether  definite  or  estimated,  are  met  by  finding  the  amount  of 
an  annuity  by  the  method  "  by  step,"  fully  described  in 
Chapters  XVI  and  XXVII,  both  of  which  contain  a  description 
of  the  longer  as  well  as  the  simplified  method  of  such  calcula- 
tion. 

The  following  Statement,  XXVI.  H.,  shows  the  method  of 
proving  the  previous  results  by  the  annvial  increment  (balance 
of  loan)  method. 


312    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


The  Redemption  Period  and  Statement  XXVI.  A. 

The  Rate  per  cent. 

The  Deductive  Method. 

Showing  tlie  metliod  of  adjusting  the  annual  instalment  in 
consequence  of  a  variation  in  the  period  of  repayment 
accompanied  by  a  variation  in  the  rate  of  accumulation, 
the  rate  of  income  from  the  present  investments  being 
unaltered  and  being  the  same  as  the  future  rate  of 
accumulation.  If  these  rates  are  unequal  or  are  varied 
proceed  as  in  Chapter  XIX. 

Conditions  before  adjustment   (at  end  of  12th  year), 

Amount  of  original  loan  repayable  in  25  years    ...  £26,495 

Amount  in  the  fund  (at  end  of  12th  year)      £9932-74 

Present  annual  income  (previously)  received  there- 
from at  3|  per  cent,  per  annum     £347'648 

Present  annual  instalment,  to  be  accumulated  for 

13  years,  at  3  per  cent £712"826 

Present  annual  increment £1060474 

Variation  from  the  above  conditions : — 

The   period   during   which   the   loan  shall   bo   redeemed   is 

reduced  from  13  to  8  years,  and 
the  rate  of  accumulatioji  of  the  fund  is  increased  from  3  to 

3^  per  cent. 

Tlie  substituted   jiciiod  of  repayment     ...   8  years. 
The  future  rate  of  accumulatioii     3^  per  cent. 


THE  RATK  PER  CENT.  AND  PERIOD 


313 


Present  investments    (at     end     of     12tli     year) 

£9932-74 


Equivalent 

amount  of 

original  loan. 


Amount  thereof,   accumulated  for  8  years  at 

3|  per  cent.  Calculation  (XXIV)  1     £13079-53 


Present  annual  instalment 


£712-826 


Amount  thereof,   accumulated  for  8  years  at 

31  per  cent.  Calculation  (XXVI)  1       £6452-28 

Provision  already  made  will,  at  the  end  of  8  years, 

repay  loan  of    £19531-81 

Additional  annual  instalment  required  :  - 

Balance,  being  amount  of  original  loan  un- 
provided for  owing  to  the  above  reduction 
in  the  period  of  repayment  from  13  to 
8  years,  but  reduced  in  consequence  of  the 
increase  in  the  rate  of  accumulation  from 
3  to  31  per  cent.,  requiring  an  additional 
annual  instalment  to  be  set  aside  and 
accumulated  for  8  years  at  3^  per  cent.  ...       £6963-19 


Additional  annual  instalment 

Calculation  (XXVI)  2     £769-270 


Amount  of  original  loan  ... 


£2649500 


Amended  annual  increment,  being  :— 

Annual  income  from  investments  .. 
Amended  annual  instalment 


£347-648 
1482-096 

£1829-744 


314  REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


The  Redemption  Period  and  Statement  XXVI.  B. 

The  Rate  per  cent. 

Showing  thp:  final  repayment  of  the  loan,  by  the  operation 
of  tlie  sinking  fund,  after  making  the  adjustment  in  the 
annual  instalment  consequent  upon  a  variation  in  the 
period  of  repayment  accompanied  by  a  variation  in  the 
rate  of  accumulation. 


Equivalent 

amount  of 

original  loan. 

Present  investments  (at  end  of  12th  vear) £99;j2'T4 


Amended  annual  increment : — 

Present  annual  instalment     i^T12'826 

Additional  annual  instalment       ...         769270 


Total  out  of  revenue     £1482-096 

Income  from  investments       347'648 


£1829-744 


Amount  thereof,   accumulated   for  8  years   at 

3i  per  cent.  Calculation  (XXIVj  6     £16562-26 


Amount  of  orin-iual    loan £26495-00 


Amended  annual  instalment         ...  £1482096 


THE    RATE    PER    CENT.    AND    PERIOD 


315 


The  Redemption  Period  and 
The  Rate  per  cent. 


Calculation  XXVI.  C. 


The  Annual  Increment  (ratio)  Method. 

To  iiud  the  amended  aunual  increment  (and  therefrom  the 
amended  annual  instalment)  in  a  sinking  fund,  in  which 
there  is  a  variation  in  the  period  of  repayment  accompanied 
by  a  variation  in  the  rate  of  accumidation,  with  or  without 
any  variation  in  the  rate  of  income  upon  the  present 
investments. 

liequired  the  annual  increment  to  be  accumulated  for  a  period 
of  8  years  at  o-|  per  cent.,  which  is  equivalent  to  an  annual 
increment  of  £1060-474,  to  be  accumulated  for  a  period  of 
lo  years  at  o  per  cent. 

(    Amount  of  £1  per  ann.,  13  years,  3%     ) 

1060-474  \ T^T-H^^^'^'"^^ 

)    Amount  of  ,£1  per  ann.,  8  years,  o-| %     j 

or  by  Table  III,  giving  the  amounts  of  £1  per  annum 

1060-474  X  15-61779 


9-05168 


Log.    Present  annual  increment 
add  Log.    Amount  of  £1  per  annum 
Table  111,  lo  years,  o  per  cent. 


=  1829-744 
1060-474 
15-61779 


3-0255000 
1-1936196 


deduct  Log.  Amount  of  £1  per  annum 
Table  III,  8  years,  3i  per  cent. 


Loff.    Amended  annual  increment 


16562-26 
905169 


4-2191196 
0-9567296 


Amended  annual  increment  ... 
To  find  the  ainendcd  animal  instalinent :  — 
deduct    the    income    from    investments, 
per  cent.         


31 


Amended  annual  instalment 

being  Present  annual  instalment...     712-826 
Additional  annual  instalment     769-270 


3-2623900 


1829-744 


347-648 


1482-096 


1482-096 


3i6  REPAYMENT   OF   LOCAL   AND    OTHER   LOANvS 


The  Redemption  Period  and  Statement  XXVI.  D. 

The  Rate  per  cent. 

The  Annual  Increment  (ratio)  Method. 

Enquired  the  annual  increment  to  be  accumulated  for  a  period 
of  8  years  at  3  per  cent,,  whicli  is  equivalent  to  an  annual 
increment  of  £1060'4T4,  to  be  accumulated  for  a  period  of 
13  years,  also  at  o  per  cent. 

To  show  the  separate  eii'ect  of  the  variation  in  the  period. 

Present  annual  increment         1060474         30255000 

multiply   by   amount   of   £1   per 

annum,   13  years,  3  per  cent.       15"61779         11936196 


16562-26        4-2191196 


divide     by    amount    of    £1    per 

annum,  8  years,  3  per  cent.  ...  8*89234 


Log.  Amended  annual  increment 

Amended  annual  increment 

beiny  Income  from  investments  ...  347*648 
Present  annual  iiislalment  ...  712'826 
Additional  annual  instalment     802058 


0-9490159 


3-2701031 


1862-532 


1862-532 


THE    RATE    PER    CENT.    AND    PERIOD 


317 


The  Redemption  Period  and 
The  Rate  per  cent. 


Statement  XXVI.  E. 


The  Annual  Increment  (ratio)  Method. 

Required  the  annual  increment  to  be  accumulated  for  a  period 
of  8  years  at  3|  per  cent.,  which  is  equivalent  to  an  annual 
increment  of  £1862-532,  to  be  accumulated  for  a  like  period 
of  8  years,  but  at  -3  per  cent. 


To  show  the  separate  effect  of  the  variation  in  the  rate  per  cent. 

Present  annual  increment         1862'532         3"270103T 

muUiphj    htj   amount    of   £1   per 
annum,  8  years,  3  per  cent.  ...  8-89234         0-9490159 


16562-26        4-2191196 


divide    hy    amount    of    £1    per 

annum,  8  years,  3^  per  cent.  ...         905169 


0-9567296 


Lop^,  amended  annual  increment  3-2623900 

Amended  annual  increment 1829-744 


heinn  Income  from  investments 


mg 


347-648 


Present  annual  instalment...     712-826 
Additional  annual  instalment     769270 


1829-744 


3i8    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


The  Redemption  Period  and  Statement  XXVI.  F, 

The  Rate  per  cent. 


The  Annual  Increment  (ratio)  Method. 

Eequired  tlie  annual  instalment  to  be  accumulated  for  a  period 
of  8  years  at  3  per  cent.,  which  is  equivalent  to  an  annual 
instalment  (as  in  XXIY.  A.)  of  £801-862,  to  be  accumulated 
for  a  like  period  of  8  years,  but  at  3|  per  cent. 

Present  annual  instalment         80r862         2-9040988 

vniltipjy    hij   amount   of   £1    per 

annum,  8  years,  3^  per  cent.  ...         9-05169         0-9567296 


7258-21        3-8608284 


divide    hy    amount    of    £1    per 

annum,  8  years,  3  per  cent.  ...         8-89234         0-9490159 


Loff.  amended  annual  instalment 


2-9118125 


Amended   annual    inslalment 


816-232 


THE    RATE    PER    CENT.    AND    PERIOD  319 


The  Repayment  Period  and  Statement  XXVI,  G, 

The  Rate  per  cent. 


The  Deductive  Method. 

Showing  for  purpose  of  proof  only  the  surplus  wliicli  will  arise 
in  the  fund  by  adopting  the  additional  annual  instalment 
of  £816-232  found  in  Statement  XXVI.  F.,  instead  of  the 
instalment  of  £802-058  in  Statement  XXYI.  D.  The 
correction  of  this  surplus  is  shown  below. 

Equivalent 

amount  of 

original  loan. 

Present  investments £9932-74 


Present  annual  increment £1060-474 


Amount  thereof,  accumulated   for  8  years  at 

3  per  cent.  Calculation  (XXYI)  3       £9430-09 

Amended  annual  instalment, 

as  in  XXVI.  F.     £816-232 


Amount  thereof,   accumulated  for  8  years  at 

3  per  cent.  Calculation  (XXVI)  4       £7258-21 


Amount  in   the   fund  at  end  of  8  years        £26621-04 

Amount  of  original  loan 26495-00 


Surplus £12604 

Annual  instalment    to  provide  £12604  at  the  end 
of  8  years  at  -\  per  cent. 

Calculation  (XXVI)  5         £14-174 

being    the    annual    instalment,     as 

shown  in  Statement  XXVI.  F.  . . .     £816-232 

less  the  annual  instalment,  as  shown 

in  Statement   XXVI.  D £802-058 

£14174 


320    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


The  Repayment  Period  and 
The  Rate  per  cent. 


Statement  XXVI .  H. 


The  Annual  Increment  (balance  of  loan)  Method. 

To  find  tlie  amended  annual  sinking  fund  instahnent  consequent 
upon  a  variation  in  tlie  period  of  repayment,  accompanied 
by  a  variation  in  the  rate  of  accumulation. 

For  Rule,  see  Chapter  XXII. 


Amount  of  original  loan    £2649500 

deduct  amount  in  the  fund  at  the  end  of  the 

12th  year £9932-74 


Balance  of  loan 


...  £16562-26 


Amended  annual  increment,  to  be  added  to  the 
fund,  and  accumulated  at  3^  per  cent.,  to 
provide  this  amount  at  the  end  of  8  years 

Calculation  (XXIY)  6     £1829-744 
deduct  income  to  be  received  from  the  present 

investments,  £9932-74,  at  31  per  cent.     ...       £347*648 


Amended    annual    instalment  £1482090 

heing  Present  annual  instalment  ...     £712-826 
Additional  annual  instalment         769270 

£1482-096 


THE    RATE    PER    CENT.    AND    PERIOD 


321 


Pro  forma  Sinking  Fund  Account  No.  11. 

A  Variation   in  the  Redemption  Period,   and  in  the   Rate   of 

Accumulation. 

Loan  of  £26,495,  repayable  at  the  end  of  25  years. 

Showing  the  final  repayment  of  the  loan,  by  the  operation 
of  the  increased  annual  instalment  of  £1482'096. 


Statement  XXYI.  B.  Rate  of  accumulation,  3^  per  cent. 


Year. 

Amount              I 
in  the  fund 
at  beginning 
of  year. 

nconie  received 
from 
investments 
Zk  per  cent. 

Annual 
sinking  fund 
instalments. 

Amount 

in  the  fund 

at  end 

of  year. 

Year. 

1 

1 

2 

2 

3 

3 

4 

The  amount  in  the 

fund  at  the 

end  of 

4 

5 

the  12th 

year,  £9932- 744,  is  the 

correct 

5 

6 

calculated  amount,  ; 

as  shown  by  C 

^alcula- 

6 

7 

tion    (XY)  2,    and 

by    the    pro 

forma 

7 

8 

account, 

No.  1,  Chapter  XV. 

8 

9 

9 

10 

10 

11 

11 

12 

9932-744 

12 

13 

9932-744 

347-648 

1482096 

11762-488 

13 

14 

11762-488 

411-687 

1482^096 

13656-271 

14 

15 

13656-271 

477-969 

1482-096 

15616-336 

15 

16 

15616-336 

546-572 

1482-096 

17645-004 

16 

IT 

17645-004 

617-575 

1482-096 

19744-675 

17 

18 

19744-675 

691-063 

1482-096 

21917-834 

18 

19 

21917-834 

767-124 

1482-096 

24167-054 

19 

20 

24167054 

845-850 

1482-096 

26495-000 

20 

21 

21 

22 

22 

23 

23 

24 

24 

25 

25 

322         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


CHAPTER  XXVII. 

SINKING    FUND    PEOBLEMS,     RELATING     TO  THE 

RATE     PER     CENT.     OF     INCOME     UPON  THE 

PRESENT  INVESTMENTS  REPRESENTING  THE 
FUND   [in  continuation  of  Chapter  XX). 

In  which  the  rate  of  income  yielded  by  such  investments 

IS  NOT  uniform  during  THE  WHOLE  OF  THE  UNEXPIRED 
PORTION    OF   THE    REPAYMENT    PERIOD. 

A.  In    WHICH    THE    FUTURE    VARIATION    IN    THE    RATE    OF 

INCOME  IS  KNOWN,    AND  IS  DEFINITE,   BOTH  AS  TO  TIME 
AND   AMOUNT.  STATEMENT    XXVII.    A. 

B,  In    WHICH    THE    FUTURE    VARIATION    IN    THE    RATE    OF 

income  is  anticipated,  but  is  uncertain,  both  as 
to  time  and  amount.  statement  xxvii.  d. 

Summary  of  methods.       How  the  variation  may  arise  and 

THE  general  considerations  APPLICABLE  THERETO.  ThE 
DEDUCTIVE     METHOD.       ThE     ANNUAL     INCREMENT     (BALANCE 

OF  loan)  method.  Statement  showing  the  final  repay- 
ment OF  THE  loan  by  THE  OPERATION  OF  THE  AMENDED 
ANNUAL      INSTALMENT.         COMPARISON      WITH      VARIATION   A 

(rate  of  income),   in  Chapter  XX,  where  the  rate  is 

UNIFORM  DURING  THE  WHOLE  PERIOD.  CALCULATION  OF  THE 
EQUATED  ANNUAL  INCOME  BY  THE  ARITHMETICAL  METHOD  AND 
DEMONSTRATION  OF  THE  ERROR  INVOLVED. 


Summary  of  the  methods  of  adjustment. 

(7)   The  ded/iictive  method,  {A)  as  summarised  helow. 

Statement  XXVII.  A. 

{II)  The  direct  method,  without  calcidation,  as  summarised 
at  the  head  of  Chapter  XX,  will  not  apply. 

(Ill)  The  annual  increment  (balance  of  loan)  mrfJiod,  as 
summarised  at  the  head  of  Chapter  XXII,  will  ap/dij  after 
finding  the  equated  annual  income  by  the  drduetive 
method  (/?),  suiiimaiised  below.  Statement  XXVII.  D. 


THE    RATE    OF    INCOME  323 

(/!')  The  annual  increment  [ratio)  method^  as  summarised 
at  the  heads  of  Chapters  XXIII,  XXV,  and  XXVI,  will  not 
apply,  as  there  is  not  any  variation  in  the  rate  of  accumulation. 

SuMMAUY  OF  THE  DEDUCTIVE  METHOD,  (A)  of  ascertaining  the 
future  or  amended  annual  sinking  fund,  instalment  due  to  a 
variation  in  the  rate  of  income  to  he  received  upon  the  present 
investments  representing  the  fund  when  it  is  Jcnoivn  at  the  time 
of  making  the  adjustment  that  such  future  rate  of  income 
will  not  he  uniform  during  the  unexpired  period  of  repayment, 
hut  will  he  varied  by  a  definite  amount  at  a  hno^vn  futiire  date. 
In  this  case  the  pioblem  is  not  complicated  by  any  variation  in 
the  rate  of  accumulation  or  the  period  of  repayment. 

Statement  XXVII.  A. 

The  terms  used  in  the  following  stimmary  ai'e  fully  cavplaincd 
at  the  head  of  Chapter  XXII .  The  unexpired  period  of  repay- 
ment is  divided  into  two  known  parts,  namely  :  — 

The  first  period,  during  which  the  rate  of  income  upon  the 
present  investments  tvill  remain  unaltered. 

The  second  period,  during  which  the  rate  of  irucoine  upon 
the  present  investments  will  he  varied  by  a  knotvn 
a,  mount. 

Memo.  (A).  If  the  rate  of  income  he  varied  at  the  time  of 
making  the  adjustment,  as  ivell  as  at  a  known  future  date, 
adjust  the  annual  instalment,  as  described  in  Chapter  XX, 
before  operation  (6)  following . 

(i)  Having  ascertained  the  value  of  the  present  investments 
in  the  manner  already  described,  calculate  the  annual 
amount  of  income  (at  each  rate  per  cent.)  to  be  received' 
during  the  first  and  second  periods  respectively . 

[2)  Calculate  the  amount  of  an  annuity,  equal  to  the  annual 

income  to  he  received  during  the  first  period  at  the 
original  rate  of  income,  for  the  number  of  years  in  that 
period,  at  the  future  rate  of  accumulation. 

Calculation  (XXVII)  1. 

(3)  Calculate  the  sum  to  which  the  amotint  so  found  in  (2) 

will  accumulate  at  the  end  of  the  number  of  years  m  the 
second  period  at  the  future  rate  nf  accumulation. 

Calculation  (XXVII)  2. 


324         REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 

(4)  Calculate  the  amount  of  an  annuity,  equal  to  the  annual 
income  to  he  received  during  the  second  period  at  the 
reduced  rate  of  income,  for  the  7iumber  of  years  in  that 
period  at  the  future  rate  of  accumulation. 

Calculation  {XXVII)  3. 

(<5)   The  amount  found  in  [3)  added  to  the  amoiint  found  in 
(4)  will  give  the  accumulated  amount  of  the  income  from 
investments  at  the  end  of  the  unexpired  period  of  repay- 
ment, expressed  in  terms  of  original  loan. 
[Here  refer  to  Memo.  A,  ahove.^ 

(6)  Calculate    the    accumulated    amount    of    the    original    or 

amended  annual  instalment  for  the  total  number  of 
years  in  the  unexpired  repayment  period  at  the  future 
rate  of  accumulation,  expressed  in  terms  of  original 
loan.  Calculation  [XIX)  2. 

(7)  Add  together  the  amounts  foiind  in  (5)  and  (6)  and  the 

value  of  the  present  investments  found  in  {1),  and  deduct 
the  total  from  the  amount  of  the  original  loan. 

{8)  The  difference  will  he  the  amount  of  loan  unpn^ovided  for 
in  consequence  of  the  above  decrease  in  the  rate  of 
income  upon  the  present  investments  during  the  second 
period. 

{9)  Calculate  the  annual  instalment  which  will  provide  the 
amount  of  loan  found  in  [8)  at  the  end  of  the  total 
unexpired  portion  of  the  repayment  period  at  the  future 
rate  of  accumulation.  Calculation  (XXVII)  5. 

[10)  The  annual  iiistalment  found  in  (9)  added  to  the  original 

or  amended  annual  instalment  foxind  in  (6)  will  be  the 
future  or  amended  annual  instalment  required. 

[11)  Prepare  a  statement  showing  the  final  repayment  of  the 

loan  by  the  operation  of  the  fund  under  the  amended 
conditions.  Statement  XXVII .  B. 

[12)  Prepare  the  usual  pro  forma  account  previously  recom- 

Tnended.  Pro  form  Account,  No.  12. 

If  the  above  problem  be  complicated  by  a  variation  in  the 
rate  of  accunfiulation,  or  the  period  of  repayment,  or  both,  it 
may  be  solved  by  a  combination  of  the  methods  previously 
(Iciiionsf rated ,   hut   irhuli   need  not.  be  specially  described. 


THE    RATE    OF    INCOME  325 

SuMMAliY    OF    THE    DeDUCTIVE    MeTHOD     (Bj     of    (ISCe rUilnDUJ 

the  future  equated  (Uinual  income  tiijon  the  yrese^it  investments 
representing  the  fund  when  it  is  antici'pated  that  the  future  rate 
of  income  will  not  he  uniform  during  the  unexpired  period  of 
repayment,  hut  the  amount  of  the  variation,  and  the  date  at 
ivhich  it  will  occur,  are  not  knoivn  at  the  time  of  maJdng  the 
adjustment. 

In  this  case  the  problem  is  not  complicated  by  any  variation 
in  the  rate  of  accumulation,  or  the  period  of  repayment. 

The  terms  used  in  the  following  summary  are  fully  ex- 
plained at  the  head  of  Chapter  XXII.  The  unexpired  period 
of  repayTnent  is  divided  into  two  estimated  parts,  as  follows  :  — 

Hie  first  period,  during  which  the  present  rate  of  income 
will  continue  to  be  received. 

The  second  period,  during  which  the  rate  of  income  is 
expected  to  be  varied,  but  the  exact  amount  of  such 
variation  can  only  be  estimated, 

(i)  Estimate,  as  accurately  as  possible,  the  period  during 
which  the  present  investments  will  continue  to  yield  the 
rate  of  income  now  received]  {The  first  period.) 

[2)  Deduct    the    number    of   years,    so    estimated,    from   the 

unexpired  portion  of  the  original  repayment  period. 

[The  second  period.) 

[3)  Estimate  as  accurately  as  possible,    the  rate  of  income 

which  will  be  yielded  by  the  present  investments  during 
the  second  period,  as  ascertained^  ni  (2) . 
{4)  Ascertain   the   value    of   the   present  investments    in    the 
manner  already  described. 

(5)  Calculate  the  annual  amount  of  income  to   be  received 

during  the  first  period,  estimated  as  in  {!),  at  the  present 
unaltered  rate  of  income. 

(6)  Calculate  the  annual  amount  of  income  expected  to   be 

received  during  the  second  period,  as  ascertained  in  [2), 
at  the  rate  per  cent.,  estimated  as  in  {3). 

(7)  Calculate  the  accumulated  amount  of  an  annuity  equal 

to  the  annual  income  to  be  received  duririg  the  first 
period  as  ascertained  in  {S)  for  the  7iumber  of  years  rn 
the  first  period  as  estimated  in  [l]  at  the  present  un- 
altered rate  of  accumulation.         Calculation  {XXVII)  I. 


326    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

(8)  Calculate  the  sum  to  tvhich  the  amount  found  in  (7)  will 

accumulate  at  the  end  of  the  second  period  found  in  [2) 
at  the  inesent  unaltered  rate  of  accuvudation. 

Calculation  (XXVIl)  2. 

(9)  Calculate  the  accumulated  amount  of  an  annuity  equal 

to  the  annual  income  estimated  to  he  received^  as  found 
in  [6)  during  the  second  period  found  in  (2)  at  the  rate 
per  cent,  of  income  estimated  in  (3)  at  the  present  un- 
altered rate  of  accumulation.  Calculation  {XX]  11)  3. 

[10)  The  amount  found  in  (8)  added  to  the  amount  found  in 

{9)  will  represent  the  amount  of  original  loan  which  will 
he  provided  at  the  end  of  the  unexpired  period  of  repay- 
ment by  the  future  accumtilation,  at  the  unaltered  rate, 
of  the  annual  amounts  of  income  from  the  present  invest- 
ments found  as  above, 

as  to  the  first  period,  in  (<5). 
as  to  the  second  period,  in  [6). 

[11)  Calculate    in   the    inanner   already    described,    using    the 

author's  standard  calculation  form,  Xo.  3x^  Chapter  X^ 
the  equal  annual  instalment  or  annuity  which  loill 
amount  to  the  total  sum  found  in  (10)  at  the  end  of  the 
unexpired  repayment  period,  at  the  unaltered  rate  of 
accumulation.  Calculation   (^Y^T  //)   6. 

(12)  The  annuity,  or  equal  animal  sum,  found  in  [11)  is  the 
equated  annual  income  required,  and  may  be  treated  as 
part  of  the  future  or  amended  annual  increment  in  all 
problems  involving  a  variation  in  the  rate  per  cent,  of 
income  upon  the  present  investments  accompanied  by  a 
variation  in  the  rate  of  accumulation. 

Fro  forma  Account,  Xo.  13. 

General  Considerations.  Reference  lias  already  been 
made  in  previous  chapters  to  the  difficulty  which  arises, 
especially  in  cases  where  the  repayment  of  the  loan  is  spread 
over  long  periods,  of  fixing  the  future  rate  of  accumulation  of 
the  sinking  fund,  and  a  similar  difficulty  will  also  occur  in 
connection  with  the  future  rate  of  income  to  be  received  upon 
the  present  investments  representing  the  amount  in  the  fund. 
Tn  adjustments  similar  to  those  under  review  the  future  rate  of 
accumulation  will  nearly  always  be  a  matter  of  speculation 
and  any  uncertainty  in  the  matter  is  met  in  practice  by  assum- 
ing a  rate  of  accumulation  on  the  low  side.     The  rate  of  income 


THE    RATE    OF    INCOME  337 

to  be  received  iu  future  upon  the  investments  representing  the 
fund  at  tlie  time  of  making'  tlie  adjustment  niay  in  some  cases 
be  assured  for  the  whole  of  the  unexpired  portion  of  the  repay- 
ment period,  and  in  Chapters  XIX,  XX,  and  XXI,  dealing 
with  Variations  A,  B,  and  C,  it  has  been  assumed  that  this  will 
be  the  case  in  order  to  simplify  the  calculation  and  to  demons- 
trate the  principle.  It  has  in  fact  been  assumed  that  the 
reduction  in  the  rate  of  income  on  the  present  investments  in 
Variations  B  and  C  has  been  partly  the  cause  of  the  rectification 
of  the  annual  instalment.  If  at  any  future  time  the  rate  of 
income  from  the  present  investments  should  again  be  reduced 
it  would  be  necessary  to  repeat  the  adjustment.  This  reduction 
in  the  rate  of  income  yielded  by  the  present  investments  may 
be  due  to  a  decrease  in  the  rate  of  interest  upon  a  security 
similar  to  a  mortgage  without  any  fall  in  the  capital  value  of 
the  investment,  or  might  be  due  to  the  realisation  of  part  of 
the  security  at  a  loss,  in  which  case  the  additional  annual 
instalment  would  inckide  the  replacement  of  the  deficiency  of 
capital,  and  a  further  amount  due  to  the  reduced  income  upon 
such  capital  realised,  although  the  actual  rate  per  cent,  yielded 
on  the  re-investment  might  remain  the  same.  But  the  rate  of 
income  to  be  received  from  the  present  investments  may  be 
reduced  at  the  time  of  making  the  adjustment,  and  at  the  same 
time  it  may  also  be  provided  that  a  further  additional  reduction 
shall  take  place  at  a  fixed  future  date.  These  are  definite  data 
which  may  be  made  the  subject  of  actual  calculation.  Such  an 
instance  occurred  in  1888,  when,  by  Mr.  Groschen's  Finance  Act, 
the  rate  of  interest  on  Consols  was  reduced  from  3  per  cent,  to 
2f  per  cent,  for  a  period  of  15  years  until  190-3,  and  the  Act 
provided  that  the  interest  should  be  then  further  reduced  to  2j 
per  cent.,  the  present  rate.  If  the  typical  Sinking  Fund  which 
has  been  used  to  illustrate  the  previous  examples  had  been,  in 
1888,  invested  in  Consols  and  had  then  an  unexpired  period  of 
13  years  to  run,  the  method  of  calculation  adopted  in  all  the 
variations  already  considered  would  have  been  accurate,  and  it 
would  have  been  quite  correct  to  base  the  additional  annual 
instalment  on  an  assured  yield  of  2|  per  cent.  But  if  the  fund 
had,  in  1888,  been  invested  in  Consols,  and  had  then  an 
unexpired  period  of  20  years  to  run,  the  problem  would  have 
been  very  different  seeing  that  the  present  investments  would 
yield  2|  pe-r  cent,  for  15  years  and  2|  per  cent,  for  the  remain- 
ing 5  years. 

A   similar   calculation   "  by   step "   has   already  been   made 
when  dealing  with  the  adjustment  of  a  deficiency  in  the  fund 


328    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

by  means  of  au  additional  annual  instalment  to  be  spread  over 
the  earlier  years  only  of  the  unexpired  repayment  period  (see 
Variation  II  (Deficiency),  Chapter  XVIj,  and  a  similar  pro- 
cedure will  apply  to  the  above  conditions.  At  the  end  of  this 
chapter  the  method  of  ascertaining  the  amount  of  an  annuity 
in  this  way  will  be  further  esjjlained  and  illustrated  by  a 
shorter  mode  of  calculation.  (Statement  XXYII.  C. 

Illustration  of  the  Method.  The  method  of  making  the 
adjustment  will  be  illustrated  by  means  of  the  results  obtained 
in  Chapters  XIX  and  XX.  In  Variation  A,  Chapter  XIX,  the 
future  rate  of  income  upon  the  present  investments  is  assumed 
to  be  I3|  per  cent,  for  the  whole  of  the  unexpired  period  of  13 
years,  and  in  Variation  B,  Chapter  XX,  to  be  reduced  to  3  per 
cent,  for  the  same  unexpired  term.  This  reduction  in  the  rate 
of  income  in  Variation  B  is  assumed  to  take  place  at  the  end  of 
the  12th  year  and  to  continue  unaltered  during  the  remaining 
13  years,  but  a  similar  change  in  the  rate  of  income  to  that  in 
the  case  of  Consols  already  referred  to  might  take  place  during 
the  unexpired  term  of  13  years  in  Variation  A.  The  present 
annual  increment  in  Variation  A  includes  income  at  3^  per 
cent,  on  investments  valued  at  £9932"74,  viz.,  £347"648  per 
annum  which,  at  the  end  of  the  unexpired  period  of  13  years, 
will  amount  at  3  per  cent.,  as  shown  by  Calculation  (XIX)  1, 
to  £5429-494. 

In  Variation  B  the  present  annual  increment  includes 
income  at  3  per  cent,  on  the  same  investments,  viz.,  £297984 
per  annum,  and  this  at  the  end  of  the  period  of  13  years  will 
amount  at  3  per  cent.,  as  shown  by  Calculation  (XX)  1,  to 
£4653-85. 

Both  the  above  annual  amounts  of  income  are  assumed  to 
accumulate  at  3  per  cent,  for  the  13  years  so  that  the  question 
of  the  rate  of  accumulation  does  not  affect  the  problem.  But 
if  in  Variation  A  there  had  been  a  reduction  in  the  rate  of 
income  taking  place  at  the  end  of  the  8th  year  of  the  unexpired 
period  of  13  years,  the  accumulated  amount  of  the  annual 
income  at  tlie  end  of  the  13  years  would  have  been  different. 

Instead  of  £347-648  per  annum  at  3|  per  cent,  for  13  years 
there  would  have  been  :  — 

Income  at  3|  per  cent.,  or  £347-648  per  annum  for  8  years, 

followed  by 
Income  at  3  per  cent.,  or  £297*984  per  onnum  for  5  years, 

both  accumulating  at  3  per  cent,  for  the  above  periods;  and  in 


THE    RATE    OF    INCOME  329 

additiou  the  accumulation  at  3  per  cent,  of  a  sum  (to  which 
£347' 648  per  auuum  will  amount  at  3  per  cent,  at  the  end  of 
8  years)  continued  without  further  annual  additiou  for  a  period 
of  5  years. 

The  amount  of  loan  which  will  be  provided  at  the  end  of 
the  period  of  13  years  by  the  accumulation  of  the  above  income 
from  investments  may  be  ascertained  by  the  following  method 
by  "  step." 

Amount    of    £347 '648    per    annum    for    8    years 
accumulated  at  3  per  cent. 

Calculation  (XXVII)  1       £3091-403 


Amount  of  the  above  sum  of  £3091'403  in  5  years 
accumulated  at  3  per  cent 

Calculation  (XXVII)  2       £3583-783 
Amount    of    £297-984    per    annum    for    5    years 
accumulated  at  3  per  cent 

Calculation  (XXVII)  3       £1582-037 


Accumulated  amount  at  the  end  of  13  years     ...       £5165-820 


as  compared  with  the  following  amounts  already  ascertained 
on  the  assumption  that  tbe  rate  of  income  will  be  uniform 
during  the  whole  period  of  13  years  :  — 

at  3i  per  cent XIX.  A.     £5429-494 

at  3    percent XX.  A.     £4653850 

The  above  sum  of  £5165-82  represents  the  portion  of 
original  loan  which  will  be  provided  at  the  end  of  the  period 
of  13  years  by  the  accumulation  at  3  per  cent,  of  the  income 
from  investments  (at  3^  per  cent,  for  the  first  8  years  and  at 
3  per  cent,  for  the  remaining  5  years). 

The  above  figures  show  the  deficiency  in  the  amount  of 
original  loan  to  be  provided  by  the  accumulation  of  the  annual 
income  from  investments  if  such  investments  had  yielded  the 
above  definite  although  variable  rates  during  the  period  of  13 
years,  as  compared  with  a  uniform  rate  of  3^  per  cent,  as 
assumed  in  the  calculation  of  the  amended  annual  instalment 
in  Variation  A,  Chapter  XIX.  The  following  Statement, 
XXVII.  A,  shows  the  deductive  method  of  ascertaining  the 
amended  annual  instalment  in  consequence  of  a  reduction  in 
the  rate  of  income  of  the  above  character. 


330  REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 

The  original  annual  instalment  as  shown  in  State- 
ment XIX.  A.  is £712-826 

and  the  additional  annual  instalment  due  to  the 
variation  in  the  rate  of  income  from  the  present 
investments  now  under  review,  as  found  by 
Calculation  (XXVII)  5,  is £16-883 


or    an    amended    annual    instalment    as    shown    by 

Statement  XXYII.  A.,  of ].     £729-709 

The  above  reduction  in  the  rate  of  income  from  the  present 
investments  at  the  end  of  the  8th  year  involves  a  further 
deficiency  of  £263-67  in  the  amount  of  loan  which  will  be 
provided  at  the  end  of  the  unexpired  repayment  period  of  13 
years,  and  is  the  difference  between  the  future  accumulation  of 
the  income  from  investments  shown  in 

Statement  XIX.  A £5429-49 

and  the  amount  ascertained  as  above     £5165-82 


£263-67 


requiring  a  further  additional  annual  instalment  of  £16-883  as 
shown  by  Calculation  (XXVII)  5. 

The  Future  Equated  Annual  Income.  The  future  or 
amended  annual  increment  will  now  be  considered.  Statement 
XIX.  A.,  Variation  A,  shows  that  the  future  or  amended  annual 
increment  to  be  accumulated  for  13  years  at  3  per  cent,  is 
£1060-474.     This  is  made  up  of  :  — 

Amended  annual  instalment       £712826 

and  future  annual  income  from  investments  at  3^ 

percent £347-648 


£1060-474 


In  the  case  now  under  consideration  the  future  annual 
increment  will  still  be  £1060-474  seeing  that  there  is  not  any 
variation  in  the  rate  of  accumulation,  as  proved  by  the  results 
obtained  in  Chapter  XX,  Variation  B,  but  it  will  be  an  equated 
and  not  an  actual  annual  increment,  ascertained  as  follows  :  — 


THE    RATE    OF    INCOME  331 

Present  annual  instalment,  as  above £712'826 

Additional  annual  instalment  due  to  the  reduction 
in  tlie  rate  of  income  from  the  present  invest- 
ments from  3^  to  3  per  cent,  during-  the  last  five 
years  of  the  period  of  repayment 

Calculation  (XXVII)  5       i>16-8S3 


Amended  annual  instalment  (Statement  XXVII.  A.)     £729" 709 
leaving  to  be  provided,  an  equated  annual  amount  of 

income  from  the   present   investments     £330'765 


Amended  annual  increment,  as  above       £1060474 


Under  the  altered  conditions,  the  actual  annual  income 
from  the  present  investments  will  be  £347 '648  per  annum  for  8 
years,  followed  by  £297"984  per  annum  for  5  years,  and  these 
annual  sums  accumulated  at  3  per  cent,  will,  at  the  end  of  the 
13  years,  amount  together  to  £5165'82.  If  the  calculation  be 
correct  the  above  equated  annual  income  (£330'765)  should 
represent  an  equated  sinking  fund  instalment  which  will 
provide  £5165"82  at  the  end  of  13  years  if  accumulated  at  3  per 
cent.,  and  this  is  found  to  be  the  case  by  Calculation  (XXVII)  6. 
The  above  amount  of  £330'765  may  therefore  be  described  as 
the  true  equated  annual  income,  being  the  annual  sum  which, 
accumulated  for  13  years  at  3  per  cent.,  is  equivalent  to  the 
two  annual  amounts  of  income  of  £347'G48  and  £297984  to  be 
accumulated  at  3  per  cent,  for  the  successive  periods  of  8  and  5 
years  as  above  described.  This  equated  annual  amount  of 
income  of  £330765  does  not  take  any  part  in  the  actual  working 
of  the  fund.  It  is  merely  the  average  annual  equivalent  (over 
the  whole  period)  of  the  known  actual  varying  amounts  which 
will  be  received  during  the  period  and  is  used  here  merely  to 
demonstrate  that  the  actual  successive  annual  amounts  of 
income  are  the  equivalents  of  the  calculated  equated  annual 
income. 

Where  the  future  variation  in  the  rate  of  income  during" 
the  unexpired  portion  of  the  repayment  period  is  definite  both 
as  to  time  and  rate  per  cent,  and  is  not  an  estimate  the  above 
deductive  method  (A)  should  be  adopted  exclusively.  This 
method  is  summarised  at  the  head  of  this  chapter. 

The  Method  by  Step.  In  Chapter  XVI,  dealing  with  the 
correction  of  a  deficiency  in  a  sinking  fund,  two  methods  of 


332    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

adjustmeut  have  been  jsliowu  depending  upon  the  period  during 
wliicli  the  additional  annual  instalment  is  required  to  be  set 
aside  and  added  to  the  fund.  In  Variation  I,  such  additional 
annual  instalment  is  spread  equally  over  the  whole  of  the 
unexpired  portion  of  the  repayment  period  of  l-J  years,  and  as 
shown  in  that  chapter  the  method  of  adjustment  is  a  simple  one. 
In  Variation  II  the  conditions  are  more  complicated  because 
it  is  required  that  the  additional  annual  instalment  shall  be 
set  aside  and  added  to  the  fund  during  the  earlier  years  only 
of  the  unexpired  portion  of  the  repayment  period.  This 
involves  an  increased  annual  charge  as  compared  with  Varia- 
tion I,  as  follows  :  — 

In  Variation  I  the  additional  annual  instalment  to 
be  spread  equally  over  the  whole  of  the  unex- 
pired repayment  period  of  13  years,  as  shown 
by  Statement  XVI.  A.,  is    £45-594 

In  Variation  II,  the  additional  annual  instalment  to 
be  set  aside  during  the  first  5  years  only  of  the 
unexpired  repayment  period  of  13  years,  has 
been  obtained  by  the  method  "  by  step  "  there 
described,  and  as  shown  in  Statement  XVI.  C,  is     £104*039 

proving  that  the  increased  annual  burden  is  due  solely  to  the 
reduction  in  the  period  allowed  for  the  adjustment  of  the 
deficiency,  the  rate  of  accumulation  being  the  same  in  both 
cases.  In  this  example  there  is  a  variation  from  the  general 
rule  applicable  to  the  accumulation  of  a  given  sum  of  money 
now  in  hand  and  also  of  an  annual  or  other  periodic  sum,  or 
annuity,  which  rule  is  based  upon  a  steady  and  uninterrupted 
accumulation,  and  does  not  provide  for  any  break  in  any  of  the 
factors  of  rate  per  cent,  or  period. 

The  present  example  relates  to  the  correction  of  a  sinking 
fund  in  consequence  of  a  variation  in  the  future  rate  of  income 
to  be  received  on  the  present  investments  representing  the  fund, 
which  occurs  in  the  middle  of  the  unexpired  portion  of  the 
repayment  period,  and  therefore  a  similar  method  by  "  step  " 
may  be  adopted. 

Having  ascertained  the  additional  annual  instalment  of 
£729'709  in  the  above  manner.  Statement  XXVII.  B.  has  been 
prepared  showing  the  final  repayment  of  the  loan.  This  state- 
ment shows  the  accumulated  amount,  £11106"10  of  the  annual 
increment  of  £1077'357  at  the  end  of  the  13  years  by  the  longer 
method   "  by   step  "   by   two   Calculations,   (XXVII)    1   and   2, 


THE    RATE    OF    INCOME  333 

relating-  to  tlie  annual  income  of  £347'648.  The  same  calcula- 
tion may  be  made  by  tbe  shorter  m^ethod  shown  in  Statement 
XXVII.  C,  described  later,  which  is  similar  to  Calculation 
XVI.  D.  1.  The  method  shown  in  Statement  XXVII.  A.  is 
the  one  which  should  be  adopted  in  cases  where  the  further 
reduction  in  the  rate  of  income  from  the  present  investments 
is  not  an  estimate,  but  is  known  and  is  definite  as  to  the  rate 
per  cent,  as  well  as  the  period.  The  difference  between  the 
arithmetical  and  true  methods  of  arriving  at  the  equated  period 
of  repayment  will  be  fully  discussed  in  Chapter  XXXII  where 
it  will  be  shown  that  the  same  principle  applies,  and  at  the  end 
of  this  chapter  the  question  of  equation  as  applied  to  the  rate 
per  cent,  wall  be  briefly  discussed. 

Calculations  (XXVII)  1  and  2  are  made  with  the  object  of 
finding  the  amount  which  will  be  provided  at  the  end  of  13 
years  by  the  accumulation  at  3  per  cent,  of  an  annuity  of 
£347' 648  to  be  set  aside  for  the  first  8  years  of  that  period  when 
the  annuity  ceases,  but  the  sum  to  which  it  then  amounts  will 
continue  to  accumulate  at  the  same  rate  for  the  remaining  5 
years. 

Calculation  (XXVII)  1  shows  that  at  the  end  of  8  years  the 
annuity  of  £347-648  will  amount  to  £3091-403,  and  Calculation 
(XXVII)  2  shows  that  this  sum  accumulated  for  a  further  5 
years  will  amount  to  £3583-783.  The  amount  of  £3091-403 
found  by  Calculation  (XXVII)  1  becomes  the  basis  of  Calcula- 
tion (XXVII)  2,  but  being  only  an  intermediate  factor  is  not 
of  any  further  interest  in  the  problem.  The  two  calculations 
may  therefore  be  combined  with  advantage  as  shown  in  State- 
ment XXVII.  C,  using  Thoman's  method,  as  being  the  simpler, 
pointing  out,  however,  that  if  the  calculation  be  required  at  any 
rate  per  cent,  not  worked  out  by  Thonian,  the  values  of  the 
factors  may  be  ascertained  by  means  of  the  formulse  already 
referred  to.  All  that  is  necessary  is  to  remember  that  (a")  of 
Thoman  may  be  found  by  means  of  the  following  factors 
referred  to  in  Chapters  IX  and  X. 

Log  a"  =  Log  RN  +  Log  ^.  _  Log  ^  j^n  _  i) 

It  is  important  to  remember,  how^ever,  that  the  method 
applies  equally  to  cases  in  which  the  rate  of  accumulation  is 
not  the  same  in  both  periods,  and  that  the  periods  in  each 
factor  EN  may  be  different.  In  the  following  Calculation 
XXVII.  C.  10  has  been  added  to  the  sum  of  the  logs  of  the 
annuity  and  of  U^,  for  the  reason  fully  explained  in  Chapter 
IX,  dealing  with  Thoman's  Tables. 


334    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

The  Arithmetical  Method  of  Finding  the  Equated 
Annual  Income.  Althougli  the  actual  amount  of  the  imme- 
diate reduction  in  the  future  rate  of  income  to  be  received  upon 
the  present  investments  may  be  known  at  the  time  of  making 
the  adjustment,  it  may  be  anticipated  that  there  will  be  a 
further  reduction  of  an  unknown  amount  at  some  future  date 
and  it  may  be  deemed  advisable  to  make  allowance  therefor. 
The  above  method  may  be  used  although  in  this  case  there  is 
one  known  and  one  estimated  rate  per  cent,  of  income. 

In  such  a  case  the  amount  of  the  further  reduction  is 
problematical,  and  it  is  therefore  preferable  and  permissible 
to  use  the  shorter  and  more  direct  arithmetical  method  of  find- 
ing the  equated  annual  income  to  be  received  over  the  period 
of  13  years,  and  this  will  now  be  shown  as  applied  to  the  above 
particulars  in  order  to  compare  the  result  with  the  mathe- 
matical method  just  described. 

There  is  (1)  an  annual  income  for  8  years  at  3|  per 

cent,   of    ". £347-648 

and  (2)  an  annual  income  for  a  further  5  years  at 

3  per  cent,  of ' £297-984 

and  it  is  required  to  find   the  equated  annual  income  for   13 
years,  which  is  equivalent  to  the  above.     Proceed  as  follows  :  — 

(1)  347-648  X  8      =   £2781-184 

(2)  297-984  x  5      -      1489-920 


£4271104 


this  total,  divided  by  13,  gives  an  annual  sum  of     £328-546 


which  is  the  arithmetical  equivalent  of  the  above  annual 
amounts  of  £347-648  and  £297-984  for  the  above  respective 
periods. 

This  calculated  annual  income  of  £328-546  upon  £9932-74, 
the  value  of  the  present  investments,  is  equivalent  to  3-31  per 
cent.,  but  the  actual  rate  per  cent,  is  immaterial.  What  is 
important  is  the  fact  that  the  estimated  annual  income  to  be 
received,  calculated  in  this  manner,  is  only  £328546,  as  com- 
pared with  the  true  equated  annual  amount  of  £'):{0-765,  ascer- 
tained by  the  mathematical  method  as  above,  or  a  decrease  of 
£2-219  per  annum. 


THE    RATE    OF    INCOME  335 

Seeing  that  tlie  total  future  or  amended  annual  increment 
tnwst  be  £1060" 474  in  order  to  provide  the  balance  of  loan 
at  the  above  rate  of  accumulation  the  estimated  annual 
deficiency  of  £2'219  must  be  added  to  the  additional  instalment 
of  £16-883  (found  as  above  in  Calculation  (XXYII)  5),  to  be 
charged  against  revenue  or  rate,  with  the  result  that  at  the  end 
of  the  period  the  fund  will  be  in  excess  of  the  proper  amount 
by  the  accumulation  of  the  larger  actual  amounts  of  the  future 
annual  income  owing  to  the  fact  that  the  actual  income  received 
will  be  in  excess  of  the  equated  amount  assumed.  Further, 
the  actual  amounts  in  the  fund  at  the  end  of  each  year  will 
exceed  the  amounts  shown  by  the  pro  forma  account  already 
referred  to,  by  an  increasing  annual  surplus,  for  the  same 
reason.  The  difference  in  this  case  is  only  small,  but  it  is  proof 
that  the  arithmetical  method  of  equation  is  incorrect,  and  the 
extent  of  the  error  depends  upon  the  actual  rates  per  cent,  of 
income,  the  periods  during  which  they  operate,  and  the  amount 
in  the  fund.  The  same  error  will  be  found  in  the  arithmetical 
method  generally  adopted  when  considering  the  equation  of  the 
period  of  repayment  in  Chapter  XXXII,  and  the  two  results 
should  be  carefully  compared. 

The  Rate  per  cent.  Statement  XXVII.  A. 

The  Deductive  Method. 

Showing  the  method  of  adjusting  the  annual  instalment,  in 
consequence  of  a  known  variation  in  the  rate  of  income 
upon  the  present  investments,  to  occur  at  a  known  future 
date,  without  any  variation  in  the  rate  of  accumulation 
or  in  the  period  of  repayment. 

The  original  conditions  in  this  example  are  similar  to 
Variation  B,  in  Chapter  XX,  in  which  case  the  reduction 
in  the  rate  of  income,  from  3^  to  3  per  cent.,  took  effect  at 
the  end  of  the  12th  year.  In  the  present  instance,  how- 
ever, the  reduction  in  the  rate  of  income,  from  3^  to  3  per 
cent.,  does  not  operate  immediately,  but  occurs  at  the  end 
of  8  years.  The  future  annual  income  from  the  present 
investments  will  therefore  be  :  — 

for  8  years,  at  3^  per  cent.,  on  £9932-74  . . .     £347648 
for  5  years  at  3  per  cent.,  on  £9932-74    ...        297984 


336         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


Present  investments  (at  end  of  12tli  year) 

Income  from  present  investments  :— 

Amount    of    £'347"648  per    annum, 
for  8  years  at  3  per  cent. 

Calculation  (XXVII)  1  £3091-403 


Amount  of  £3091-403,  in  5  years  at 
3  per  cent. 

Calculation  (XXVII)  2  £3583-783 
Amount   of   £297-984  per   annum, 
for  5  years  at  3  per  cent. 

Calculation  (XXVII)  3  £1582-037 


Present  annual  instalment  (Variation  A) :  — 

Amount    of   £712-826    per   annum, 
for  13  years  at  3  per  cent. 
Calculation  (XIX)  2  (£680-234)  £10623-75 
Calculation  (XIX)  3  (£32-592)  509-02 


Present  annual  instalment 


£712-826 


Equivalent 

amount  of 

original  loan. 

£9932-74 


£5165-82 


-    £11132-77 


Provision  already  made  will  repay  loan  of 

Additional  annual  instalment  required  :— 

Balance,  being  amount  of  original  loan  un- 
provided for,  owing  to  tlie  above  reduction 
in  the  rate  of  income  from  tbe  present 
investments  during  the  later  years  of  the 
unexpired  portion  of  the  repayment  period, 
requiring  an  additional  annual  instalment 
to  be  set  aside  and  accumulated  for  13  years 
at  3  per  cent 

Additional  annual  instalment 

Calculation  (XXVII)  5     £16-883 


£26231-33 


£263-67 


Amount  of  original  loan 

Amended  annual   instalment     £729-709 


£2649500 


THE    RATE    OF    INCOME 


337 


The  Rate  per  cent. 


Statement  XXVII.  B. 


Showing  the  final  repayment  of  the  loan,  by  the  operation 
of  the  sinking  fund,  after  making  the  adjustment  in  the 
annual  instalment,  consequent  upon  a  variation  in  the  rate 
of  income  upon  the  present  investments  to  occur  at  a  known 
future  date,  without  any  variation  in  the  rate  of  accumula- 
tion or  period  of  repayment. 


Amended  annual  increment  for  8  years  :  — 

Present  annual  instalment     ...     £729709 


Equivalent 

amount  of 

original  loan. 


Income   from   investments,    3. 
per  cent 


347-648 


£1077-357 


Amount  thereof,   accumulated  for  8  years  at 

3  per  cent.  Calculation  (XXVII)  7       £9580-22 


Amount  of  this  sum,  accumulated  for  a  further 

5  years  at  3  per  cent.    Calculation  (XXVII)  8     £11106-10 

Amended  annual  increment  for  5  years  :  — 

Annual  instalment,  as  above...     £729-709 
Income    from    investments,    3 

percent 297-984 


£1027-693 


Amount  thereof,   accumulated  for  5  years  at 

3  per  cent.  Calculation  (XXVII)  9       £5456-16 

£16562-26 

« 

Present  investments  (at  end  of  12th  year)    £9932-74 

Amount  of  original  loan £26495-00 


338 


REPAYMENT   OF    LOCAL   AND    OTHER    LOANS 


The  Rate  per  cent. 


Statement  XXVII.  C. 


The  Amount  of  {the  Avio^mt  of  £1  per  annum). 
Method  by   Step,   by  Thoman's   Tables. 

To  find  tlie  accumulated  amount  of  an  annuity  to  be  added 
to  the  sinking  fund  for  a  limited  period  of  years,  and  at 
the  end  of  that  period  the  accumulated  amount  thereof 
to  continue  to  accumulate  for  a  further  specified  period. 
The  rate  of  accumulation  in  both  cases  may  be  the  same, 
or  be  at  different  rates  per  cent. 

Eequired  the  amount  of  an  annuity  of  <£347"648  to  be  added  to 
the  sinking  fund  for  a  period  of  8  years,  and  accumulated 
at  3  per  cent.  At  the  end  of  8  years  the  annuity  ceases, 
but  the  sum  to  which  it  has  then  amounted  continues  to 
accumulate  for  a  further  period  of  5  years  at  3  per  cent. 

First  period,  8  years;  second  period,  5  years. 


Log.    annuity 

add  Log.  W^,  3  per  cent.          8  years 
Log.  RN,  3  per  cent.         5  years 


347-648 


2-5411397 
0-1026978 
0-0641861 

2-7080236 


deduct  Log.  a",  3  per  cent.  8  years 


add  10       12-7080236 
91536819 


3-554341' 


which  is  the  log.   of  the  required   future   amount 

at  the  end  of  13  years       £3583-783 


Note.     This  statement  may  be  compared   with    Statements 
XVI.  D.l  and  XXXIV.  G. 


THE    RATE    OF    INCOME 


339 


Pro  forma  Sinking  Fund  Account,  No,  12, 

A  Variation  in  tlie  Hate  of  Income  from  Investments,  which 
is  not  uniform  over  the  unexpired  Hepayment  Period. 

Loan  of  £26,495,  rejjayable  at  the  end  of  25  years. 

Showing  the  final  repayment  of  the  loan,  by  the  operation 
of  the  increased  annual  instalment  of  £729709. 

Statement  XXVII.  B.  Rate  of  accumulation,  3  per  cent. 


Year. 
1 

2 

Amount 
in  the  fund 

at 

beginning  of 

year. 

Income 
received'fiom 
investments 

made  up 
to  Vith  year. 

Annual 

sinking 

fund 

instalment. 

received  from 
investments 

made  after 
l'2th  year 

3  per  cent. 

Amount 

in  the  fund 

at 

end  of 

year. 

Year. 
1 

JL 

2 

3 

3 

4 

The 

amount  in  the  fun( 

i  at  the  end  of 

4 

5 

the  : 

I2th  year. 

£9932-744,  is  the  correct 

6 

6 

calculated  amount,  as  sh 

own  by  Calcula- 

6 

7 

tion 

(XV)  2, 

and    by 

the    pro 

forma 

7 

8 

9 

10 

account,  No.  1, 

Chapter  XV. 

8 

9 

10 

11 

11 

12 

9932-744 

12 

13 

9932-744 

347-648 

729-709 

Nil 

11010-101 

13 

14 

11010101 

347-648 

729-709 

32-320 

12119-778 

14 

15 

12119-778 

347-648 

729-709 

65-610 

13262-745 

15 

16 

13262-745 

347-648 

729-709 

99-900 

14440002 

16 

17 

14440002 

347-648 

729-709 

135-216 

15652-575 

17 

18 

15652-575 

347-648 

729-709 

171-594 

16901-526 

18 

19 

16901-526 

347-648 

729-709 

209070 

18187-953 

19 

20 

18187-953 

347-648 

729-709 

247-654 

19512-964 

20 

21 

19512-964 

297-984 

729-709 

287-407 

20828-064 

21 

22 

20828-064 

297-984 

729-709 

326-860 

22182-617 

22 

23 

22182-617 

297-984 

729-709 

367-494 

23577-804 

23 

24 

23577-804 

297-984 

729-709 

409-350 

25014-847 

24 

25 

25014-847 

297-984 

729-709 

452-460 

26495000 

25 

340         REPAYMENT    OF    LOCAL   AND    OTHER   LOANS 

The  Rate  per  cent.  Statement  XXVII.  D. 

The  Annual  Increment  (balance  of  loan)  Method. 

To  find  the  amended  annual  sinking  fund  instalment,  conse- 
quent upon  a  known  variation  in  the  rate  of  income  upon 
the  present  investments  to  occur  at  a  known  future  date, 
based  upon  the  equated  annual  income. 

The  rule  relating  to  this  method  is  stated  at  the  head 
of  Chapter  XXII. 

Amount  of  original  loan  (25  years) £2649500 

deduct  amount  in  the  fund  at  the  end  of  the 

12th  year £9932-74 


Balance  of  loan £16562-26 


Annual  increment,  to  be  added  to  the  fund,  and 
accumulated  at  3  per  cent.,  to  provide  this 
amount  at  the  end  of  13  years 

'  Calculation  (XX)  4     £1060474 
deduct    the    equated    annual    income    to    be 
received  from  the  present  investments  ascer- 
tained as  described  in  the  text     £330-765 


Amended    annual   instalment £729-709 

heing  Present  annual  instalment  ...     £712-826 
Additional      instalment,      as 
found  in 

Statement  XXVII.  A.       £16-883 

£729-709 


Note.  This  method  will  be  of  use  where  the  variation  in 
the  rate  of  income  is  of  the  above  unequal  nature,  and  is 
combined  with  a  variation  in  the  rate  of  accumulation,  as  in 
Chapter  XXI  (Variation  C). 


THE    RATE    OF    INCOME 


341 


Pro  forma  Sinking  Fund  Account,  No.  13, 

A  Variation  in  the  Rate  of  Income  from  Investments,  which 
is  not  uniform  over  the  unexpired  Repayment  Period. 


Loan  of  £26,495,  repoi/ahle  at  the  end  of  25  yearx. 

Showing  the  final  kepayment  of  the  loan,  by  the  operation 
of  the  increased  annual  instalment  of  £729709,  and  the 
equated  annual  income  of  £3-30'765. 


Statement  XXYII.  D. 


Rate  of  accumulation,  3  per  cent. 


^ear. 

Amount 

in  the  fund 

at  beginning 

of  year. 

Equated 

annual  income 

from 

investments 

after  li'th  year. 

Annual 
sinking  fund 
instalment. 

Income  to  be 

received  from 

investments 

made  after 

12th  year. 

Amount 

in  the  fund 

at  end 

of  year. 

Year. 

1 

1 

2 

2 

3 

3 

4 

The 

amount  in  the  fund  at  the  end  of 

4 

6 

the 

12th  year, 

£9932-744,  is  the  correct 

5 

6 

calculated  amount,  as  sh 

own  by  Ca 

Icula- 

6 

7 

tiou 

(XV)  2, 

and    by 

the    pro 

forma 

7 

8 

9 

10 

acco 

unt,  No.  1, 

Chapter  XV. 

8 

9 

10 

11 

11 

12 

9932-744 

12 

13 

9932-744 

330-765 

729-709 

— 

10993-218 

13 

14 

10993-218 

330-765 

729-709 

31-814 

12085-506 

14 

15 

12085-50G 

330-765 

729-709 

64-583 

13210-563 

15 

16 

13210-563 

330-765 

729-709 

98-335 

14369-372 

16 

17 

14369-372 

330-765 

729-709 

133-099 

15562-945 

17 

18 

15562-945 

330-765 

729-709 

168-906 

16792-325 

18 

19 

16792-325 

330-765 

729-709 

205-787 

18058-586 

19 

20 

18058-586 

330-765 

729-709 

243-775 

19362-835 

20 

21 

19362-835 

330-765 

729-709 

282-903 

20706-212 

21 

22 

20706-212 

330765 

729-709 

323-204 

22089-890 

22 

23 

22089-890 

330-765 

729-709 

364-714 

23515-078 

23 

24 

23515078 

330-765 

729-709 

407-470 

24983-022 

24 

25 

24983-022 

330-765 

729-709 

451-504 

26495000 

25 

Section  V. 

Sinking  Fund  Problems. 

The  Date  of  Borrowing  and  the 

Redemption    Period. 


345 


CHAPTER  XXVIII. 

SINKING  FUND  PROBLEMS  RELATING  TO  THE 
DATE  OF  BORROWING  AND  THE  REDEMPTION 
PERIOD 

Without  any  complication  as  kegakds  the  life  or  duration 

OF    continuing    utility   of   the   ASiSET    CHEATED   OUT   OF    THE 
LOAN. 

Loan  borkowed  over  several  years,  in  one  sum  in  each  year, 

EACH  year's  borrowings  BEING  REPAYABLE  IN  A  PRESCRIBED 
PERIOD  FROM  THE  DATE  OF  BORROWING. 

1.  By  MEANS  OF  ONE  SINKING  FUND  ONLY. 

2.  By    separate    sinking    funds    for    each    year's 

borrowings. 

The  foregoing  chapters  deal  with  the  various  problems 
likely  to  arise  in  connection  with  the  sinking  funds  of  local 
authorities  and  commercial  and  financial  undertakings  affecting 
(1)  the  amount  in  the  fund  at  any  time;  (2)  the  rate  per  cent, 
of  accumulation;  (3)  the  rate  per  cent,  of  income  upon  the 
present  investments  representing  the  fund;  (4)  the  period  of 
repayment;  and  (5)  various  combinations  of  the  above  factors. 
In  the  whole  of  the  examples  which  have  been  used  to  illustrate 
such  problems  it  has  been  assumed,  for  the  purpose  of 
calculating  the  original  or  amended  annual  instalment  to  be 
set  aside  and  accumulated  as  a  sinking  fund  to  provide  a  given 
loan  at  the  end  of  any  period,  that  the  loan  was  borrowed  in 
one  year  and  on  one  date,  namely,  at  the  beginning  of  the 
financial  year,  and  that  the  first  annual  instalment  was  set 
aside  at  the  end  of  that  year.  This  method  of  treating  an 
annuity  or  other  periodic  payment  is  the  basis  of  all  such 
calculations  and  upon  which  the  formulae  and  tables  are 
constructed.  This  ideal  procedure  may,  it  is  true,  be  met  with 
occasionally;  but  as  a  matter  of  fact  it  very  seldom  occurs  in 
actual  practice.  It  has  been  assumed  in  all  cases  that  it  has  so 
happened  in  order  to  simplify  the  conditions  and  to  demonstrate 
the  actuarial  principles  underlying  the  repayment  of  debt  in 
this  manner,  without  introducing  any  extraneous  complications. 
The  time  has  now  arrived  when  it  is  necessary  to  consider  the 
conditions  occurring  in  actual  practice. 

The  variations  from  the  ideal  method  of  borrowing  are  of  a 
twofold   nature   and   arise   when   the   borrowings   are   made   at 


346    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

various  dates  in  any  one  year  or  are  spread  over  several  years. 
If  the  total  loan  be  repayable  on  a  given  date  both  these 
variations  may  necessitate  an  adjustment  in  the  annual  instal- 
ment. In  the  case  of  a  loan  borrowed  at  various  dates  during 
any  one  year  the  necessity  for  any  adjustment  depends  upon 
the  magnitude  of  the  loan  and  affects  only  the  first  and  last 
years  of  the  term.  Any  neglect  to  make  such  adjustment 
cannot  prolong  the  period  of  repayment  for  more  than  part 
of  a  year,  but  in  the  case  of  borrowings  spread  over  several 
years  the  matter  becomes  more  important.  There  are  other 
factors  which  may  further  complicate  the  problem,  depending 
upon  the  nature  of  the  outlay  and  the  periods  of  repayment 
allowed  for  each  class.  In  some  cases  the  power  or  sanction 
specifies  not  only  the  total  amount  of  loan  authorised,  but  also 
gives  details  of  the  component  parts  of  such  loan  divided  as 
between  the  various  classes  of  outlay,  each  having  its  own 
period  of  repayment.  In  some  cases,  however,  the  total  amount 
authorised  is  stated  without  any  such  subdivision,  and  an 
equated  period  is  prescribed  for  the  repayment  of  the  total  loan. 

The  after  consideration  of  the  subject  will  be  divided  into 
two  parts  depending  upon  the  character  of  the  outlay,  which 
may  be  all  of  one  nature  having  a  similar  life  or  period  of 
utility  and  a  consequent  equal  period  of  repayment.  On  the 
other  hand,  the  outlay  under  one  power  or  sanction  may  consist 
of  various  classes  for  each  of  which  a  separate,  and  varying, 
period  of  repayment  is  imposed. 

It  may  be  accepted  as  a  general  rule  that  the  repayment 
period  now  allowed  by  Parliament  is  fixed  with  regard  to  the 
probable  life  or  duration  of  continuing  utility  of  the  works. 
This  variation  in  the  life  of  the  asset  imports  special  difficulties 
into  the  problem,  relating  to  the  vexed  question  of  the  adequacy 
or  otherwise  of  the  sinking  fund  instalment  as  a  provision  for 
depreciation,  obsolescence  and  supersession. 

Dealing  first  with  the  actual  borrowings,  the  subject  will  be 
treated  in  the  following  order,  namely  :  — 

I.  Loans  authorised  for  outlay  all  of  one  nature  having  the 
same  period  of  repayment. 
As  regards  the  actual  borrowing  such  loans  may  be  divided 
into  three  classes  as  follows  : 

(a)  Loan  borrowed  over  several  years,,  in  one  sum  in  each 
year,  repayable  over  a  term  of  years  in  a  prescribed 
period  from  the  several  dates  of  borrowing.  Such  loans 
will  be  described  in  this  chapter. 


THE   DATES    OF    BORROWING   AND   REPAYMENT      347 

(6)  Loan  borrowed  over  several  years,  in  one  sum  in  each 
year,  repayable  in  one  sum  on  a  certain  specified  date. 

Chapter  XXIX. 

(c)  Loan  borrowed  in  one  or  more  years,  in  varying  amounts 
at  varying  dates  in  each  year,  repayable  in  one  sum  on 
a  certain  specified  date,  where  it  is  required  that  the 
revenue  or  rate  account  of  each  year  of  borrowing  shall 
be  charged  with  a  proportionate  part  of  the  annual 
sinking  fund  instalment.  Chapter  XXX. 

II.  Loans  authorised  for  outlays  of  varying  nature,  each 
having  a  different  life  or  period  of  continuing  utility, 
the  whole  of  the  loan  to  be  repaid  on  one  uniform  date. 

Chapter  XXXII. 

It  is  not  necessary  to  do  more  than  point  out  that  if  at  any 
future  time  during  the  operation  of  the  fund  any  question 
should  arise  as  to  a  variation  in  the  repayment  period,  the  rate 
of  accumulation,  or  the  rate  of  income  to  be  received  from  the 
present  investments  representing  the  fund,  the  problem  may 
be  solved  by  one  or  other  of  the  methods  described  in  previous 
chapters. 

The  total  amount  of  any  loan  sanctioned  for  purposes  of  large 
public  works  is  rarely  required  to  be  raised  in  one  year.  The 
actual  construction  often  occupies  several  years,  and  such  an 
amount  only  is  borrowed  in  any  one  year  as  will  be  sufficient 
to  pay  for  the  works  actually  constructed  in  that  year.  The 
complication  of  the  sinking  fund  owing  to  the  loan  being 
borrowed  over  a  period  of  years  may  be  obviated  by  borrowing 
one  amount  in  advance;  but  the  results  of  borrowing  largely 
in  excess  of  the  actual  annual  requirements  are  :  — 

(1)  A  loss  of  interest  owing  to  the  money  borrowed  lying 

in  the  bank. 

(2)  The  excessive   sinking  fund   instalments  which   have   to 

be   set  aside,    and  provided   out   of  revenue  or  rate,   in 

respect  of  the  amount  borrowed  in  excess  of  the  actual 

requirements. 

The   Act   authorising   the   borrowing   generally   contains  a 

clause  limiting  the  period   of  repayment  either  to   a   definite 

number  of  years  from  the  date  of  borrowing  or  to  a  specified 

date,  and  the  limitation  applies  to  the  amount  borrowed  in  each 

year.     If  the  construction  extends  over,  say,  four  years,   this 

will  entail  four  separate  calculations  of  the  annual  instalment; 

and  the  generally  recognised  practice  is  to  treat  each  year's 


348    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

borrowings  a.s  a  separate  loan  repayable  in  the  prescribed 
ntimber  of  years  from  the  actual  date  of  borrowing  and 
requiring  a  separate  sinking  fund.  In  the  case  of  a  large 
municipality  this  entails  the  keeping  of  a  great  number  of 
sinking  fund  accounts,  and  the  annual  provision  of  the  instal- 
ments becomes  a  very  detailed  process.  Further,  the  final 
repayment  of  the  loans  is  spread  over  a  term  of  years  equal 
to  the  extended  period  of  borrowing. 

In  the  case  of  loans  raised  by  the  issue  of  stock  the  whole 
of  such  stock  is  made  redeemable  at  the  end  of  a  specified 
number  of  years,  or,  as  is  generally  the  case,  on  a  definite 
date.  In  this  case  there  need  only  be  one  sinking  fund  with 
four  separate  annual  instalments  set  aside  each  year  for 
decreasing  periods,  but  all  calculated  to  mature  at  the  same 
date  and  at  the  same  rate  of  accumulation. 

The  present  example  will  be  illustrated  by  the  sinking  fund 
fully  described  in  Chapter  XY,  relating  to  the  repayment  of  a 
loan  of  £26,495  in  25  years  at  an  accumulation  rate  of  3^  per 
cent.  In  this  case  it  has  been  ascertained  that  if  the  whole  of 
the  loan  were  borrowed  in  one  year,  namely,  at  the  beginning 
of  the  financial  year,  it  required  an  annual  instalment  of 
i:/680"234  to  be  set  aside  at  the  end  of  the  first  and  24  subse- 
quent years  in  order  to  repay  the  original  loan.  If  the  loan, 
instead  of  being  borrowed  in  one  year,  were  borrowed  over  a 
period  of  four  years,  and  the  sanction  or  authorisation  imposed 
the  period  of  redemption  of  25  years  in  respect  of  each  amount 
borrowed,  the  conditions  would  havebeen  considerably  modified. 
There  is  in  this  case  the  equivalent  of  four  separate  loans  each 
repayable  in  25  years,  but  maturing  at  the  end  of  four  successive 
years.  It  will  be  assumed  that  each  amount  of  loan  Avas 
borrowed  at  the  beginning  of  the  financial  year,  or  if  borrowed 
on  several  dates  in  that  year  that  no  necessity  exists  to  equate 
the  borrowing  at  the  various  dates.  It  will  be  further  assumed 
that  the  above  amount  of  £26,495  was  borrowed  in  unequal 
amounts  in  each  of  the  four  years,  and,  in  order  to  avoid  making 
four  separate  additional  calculations,  that  a  definite  proportion 
of  the  loan  was  borrowed  in  each  year.  Seeing  that  the  annual 
instalment  is  based  upon  the  amount  of  £1  per  annum  for 
25  years,  it  is  obvious  that  it  is  directly  proportionate  to  the 
amount  of  the  loan  and  that  the  four  annual  instalments  may 
be  found  by  dividing  the  original  annual  instalment  of 
£680'234  in  the  same  proportions  as  the  total  loan  is  divided. 

The  following  table  shows  the  actual  details  of  the  loan 
under  consideration  :  — 


THE   DATES    OF   BORROWING   AND    REPAYMENT      349 


TABLE  XXYIII.  A. 

Loan  of  £26,495,  borrowed  over  four  years.     Eepayment  spread 
over  a  similar  period. 

Original  annual  instalments  all  calculated  to  mature  in  25  years 
but  at  the  end  of  successive  years. 

Rate  of  accumulation  3|  per  cent. 


Year 

of 

borrowing. 

Redemption 
period. 

Proportion 
borrowed 
each  year. 

Amount 
borrowed 
each  year. 

Annual 
instalment 
on  yearly 
borrowing. 

Annual 
instalment 
at  end  of 
each  year. 

First 

25  vears 

^/:4 

3785- 

97-176 

97-176 

Second 

j» 

'U. 

5677-5 

145-764 

242-940 

Third 

j> 

Vi. 

7570- 

194-353 

437-293 

Fourth 

jj 

'U. 

9462-5 

242-941 

680-234 

26495- 

680-234 

— 

There  are  two  alternative  methods  of  keeping  the  sinking 
fund  accounts  in  such  a  case.  One  method  is  to  keep  one 
sinking  fund  only,  and  to  set  aside  an  increasing  instalment 
during  the  first  four  years,  a  constant  instalment  during  the 
next  21  years,  and  a  decreasing  instalment  during  the  final 
three  years  of  the  total  period  of  28  years  during  which  the 
fund  will  run.  If  this  method  be  applied  to  the  foregoing 
example,  the  annual  instalments  added  to  the  fund  will  be  as 
follows  :  — 

TABLE  XXVIII.B. 

Loan  of  £26,495,  borrowed  over  four  years.     Repayment  spread 
over  a  similar  period. 

Annual  instalments  to  be  added  to  one  sinking  fund  relating-  to 
the  total  loan. 


To  be  set  aside  at  end  of 

1st  year  

2nd  year 

3rd  year  

4th  year  

5th  to  25th  year  =  21  years 

26th  year  

27th  year  

28th  year  


Annual  instalments. 

97-176 
242-940 
437-293 
680-234 
680-234 
583-058 
437-294 
242-941 


Total. 

97-176 
242-940 
437-293 
680-234 
14,284-914 
583-058 
437-294 
242-941 


being  the  equivalent   of  25   annual  instalments 

pf  £680-234      , £17,005-850 


350    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

There  are  several  objections  to  keeping  the  sinking  fund 
accounts  in  this  manner,  all  of  which  are  practical.  The  first 
is  that  unless  a  proper  pro  forma  account  be  at  once  made  out 
showing  the  operation  of  the  fund  until  maturity  there  will  be 
a  liability  to  continue  the  full  instalment  of  £680"234  beyond 
the  25th  year.  A  further  error  may  possibly  arise  owing  to 
the  application,  during  the  period,  of  part  of  the  fund  in 
redeeming  part  of  the  debt.  As  previously  stated,  if  any  part 
of  the  fund  be  so  applied  it  is  requisite  and  obligatory  to  pay 
into  the  fund,  annually,  interest  upon  the  loan  so  redeemed  at 
a  rate  per  cent,  at  least  equal  to  the  calculated  rate  of  accumula- 
tion. Although  this  obligation  may  be  remembered  and  carried 
out  during  the  first  25  years  it  may  then  be  overlooked  that  at 
that  time  £3,785  of  loan  has  been  fully  redeemed  and  that 
interest  upon  this  amount  of  loan  repaid  need  no  longer  be 
charged  to  the  revenue  or  rate  account  and  added  to  the  fund. 
The  same  factor  of  error  may  arise  at  the  end  of  the  26th,  2Tth 
and  28th  years.  Taking  the  above  possible  sources  of  error 
into  consideration,  it  is  preferable  to  adopt  a  method  which  will 
avoid  them,  although  it  may  entail  a  little  more  clerical  work. 
The  proper  method  in  such  cases  is  to  keep  a  separate  sinking 
fund  for  each  year's  borrowings,  and  to  prepare  at  the  dates 
of  borrowing  a  pro  forma  account  showing  the  operation  of  each 
fund  until  maturity. 

It  cannot  be  too  often  repeated  that  this  pro  forma  account 
should  be  prepared  in  respect  of  every  sinking  fund.  If  the 
method  of  separate  sinking  funds  be  adopfed  it  will  ensure  that 
proper  payments  of  interest  in  respect  of  debt  redeemed  out  of 
the  fund  are  made  to  the  fund  each  year  and  will  also  enable 
arrangements  to  be  made  to  repay  each  loan  at  the  end  of  the 
prescribed  period.  It  will  also,  in  the  case  of  long  repaj-ment 
periods,  avoid  the  necessity  of  referring  to  old  ledgers  or  books 
of  account  which  may  have  been  destroyed. 

For  this  purpose  it  is  an  advantage  to  earmark  each  fund, 
and  also  the  corresponding  instalment,  in  some  such  way  as  the 
following  :  — 

Gas  Works   Sinking  Fund.     Sanction  1900. 
Loan  of  1901 — 25  Years, 

and  to  number  each  instalment. 

The  charge  to  tlie  revenue  or  rate  account  at  the  end  of  the 
third  year  would  be  made  up  as  follows  :  — 


THE   DATES    OF   BORROWING    AND    REPAYMENT       351 

Sanction  1900 — 25  Years. 

Loan  of  1901.     3rd  Instalment     £97-176 

1902.  2nd         do 145-764 

1903.  1st  do 194-353 


Totallnstalment £437-293 


At  the  end  of  each  of  the  last  four  years  one  of  the  original 
year's  borrowings  will  be  repaid ;  and  the  charge  to  revenue  or 
rate  at  the  end  of  the  26th  year  (when  the  loan  borrowed 
during  the  first  year  has  been  entirely  repaid)  will  be  as 
follows  :  — 

Sanction  1900 — 25  Years. 

Loan  of  1901.  Repaid nil 

1902.  25th  Instalment £145-764 

1903.  24th       do 194-353 

1904.  23rd        do 242-941 


Totallnstalment £583058 


Although  the  method  of  keeping  separate  sinking  funds  for 
each  year's  borrowings  under  each  sanction  or  authorisation  is 
here  advocated,  it  must  not  be  assumed  that  this  method 
requires  that  separate  bank  accounts  should  be  kept  for  each 
fund.  This  would  become  intolerable  in  practice  even  if  the 
bank  would  agree  to  do  so.  Neither  is  it  necessary  to  keep  a 
separate  investment  account  for  each  fund.  One  bank  account 
and  one  investment  account  for  each  department  of  the  local 
authority  is  quite  sufficient  because  at  the  end  of  any  year  the 
amount  in  the  bank,  the  amount  invested,  and  the  loans  repaid 
out  of  sinking  fund  should  together  be  equal  to  the  amount 
standing  to  the  credit  of  the  fund,  or  to  the  credit  of  the  whole 
of  the  funds  of  the  particular  department.  In  this  connection 
it  is  important  to  point  out  that  loans  repaid  by  means  of  the 
sinking  fund  should  be  treated  as  an  investment  of  so  much  of 
the  fund  so  applied  and  be  debited  to  a  special  account  instead 
of  being  debited  to  the  sinking  fund  account,  in  the  same  way 
that  investments  in  outside  securities  are  kept  in  separate 
accounts.  The  reason  for  doing  this  is  to  ensure  that  the 
revenue  or  rate  account  is  annually  debited,  and  the  sinking 
fund  credited,  with  the  proper  amount  of  interest  in  respect  of 
such  part  of  the  fund  so  applied  in  redemption  of  debt.     If  the 


352    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

accounts  are  kept  in  such  a  manner  tlie  sinking  fund  will,  at  the 
end  of  each  year,  show  the  amount  of  loan  provided  for  out  of 
revenue  or  rate,  and  will  enable  a  comparison  to  be  made  with 
the  pro  forma  account  already  referred  to  and  recommended. 
By  this  means  only  can  any  variation  from  the  calculated 
amount  which  sliould  be  in  the  fvmd  at  any  time  be  readily 
ascertained  and  immediately  adjusted.  This  applies  to  all 
sinking  funds. 

It  ought,  however,  to  be  pointed  out  that  if  only  one  bank 
account  and  one  investment  account  be  kept  it  will  be  necessary 
to  apportion,  as  between  the  different  sinking  funds,  the  interest 
allowed  by  the  bankers  and  the  income  received  from  invest- 
ments whether  the  investments  be  in  outside  securities  or 
consist  of  loans  redeemed  out  of  the  sinking  fund.  In  ordinary 
cases  there  may  be  some  difficulty  in  doing  this  because  the 
interest  allowed  by  the  bank  upon  balances  in  hand  will  almost 
certainly  be  at  a  lower  rate  than  the  calculated  rate  of  accumu- 
lation of  the  fund.  This  difficulty  is,  however,  removed  by  the 
fact  that  there  is  in  the  case  of  each  sinking  fund  a  standard 
to  work  to,  namely,  the  pro  forma  account  previously  prepared 
showing  the  amount  which  should  stand  to  the  credit  of  each 
fund  at  the  end  of  each  year  of  the  repayment  period.  If  the 
amount  of  income  actually  received  from  the  investment  of  the 
sinking  fund  in  outside  securities  and  in  loans  redeemed  falls 
short  of  the  amount  originally  calculated  to  be  received,  such 
deficiency  should  be  made  good  each  year  by  charging  it  to  the 
revenue  or  rate  account  and  paying  the  deficiency  into  the 
sinking  fund  bank  account.  The  necessity  to  apportion  the 
interest  allowed  by  the  bank  and  the  income  received  from 
investments  may  be  entirely  removed  by  crediting  the  interest 
allow^ed  by  the  bank,  as  well  as  the  income  received  from  the 
investments,  to  a  sinking  fund  interest  suspense  account.  The 
suspense  account  should  be  debited  with  the  total  amount  of 
interest  which  ought,  according  to  the  pro  forma  accounts,  to 
be  credited  to  the  various  sinking  funds,  and  the  balance 
remaining  to  the  debit  of  the  suspense  account,  will  show  the 
amount  of  the  deficiency  of  interest  to  be  debited  to  the  revenue 
or  rate  account.  By  this  means  not  only  will  the  amount 
standing  to  the  credit  of  the  sinking  fund  agree  each  year  with 
the  amount  which  should  so  stand  according  to  the  pro  forma 
account,  but  there  will  be  the  further  advantage  that  eacli 
year's  revenue  or  rate  account  will  bear  its  proper  burden  and 
there  will  never  arise  any  necessity  to  make  provision  for  a  large 
deficiency  in  any  sinking  fund  caused  by  an  accumulation  of 


THE   DATEvS    OF   BORROWING   AND    REPAYMENT      353 

many  annual  deficiencies  in  the  income  which  ought  to  have 
accrued  to  the  fund. 

This  method  of  keeping  separate  sinking  funds  for  eacli 
year's  borrowings  is  not  required  in  the  case  of  loans  issued 
by  way  of  a  stock  redeemable  at  a  fixed  date,  seeing  that  the 
repayment  of  the  loan  is  not  spread  over  a  number  of  years 
equal  to  the  number  of  years  occupied  by  the  borrowing. 

A  loan  borrowed  over  a  series  of  years  repayable  in  one  sum 
at  a  fixed  date  will  be  considered  in  the  next  chapter,  and,  for 
the  sake  of  comparison,  the  figures  used  in  this  example  will  be 
further  utilised. 


354         REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 


CHAPTER    XXIX. 

SINKING  FUND  PROBLEMS,  RELATING  TO  THE 
DATE  OF  BORR(JWING  AND  THE  REDEMPTION 
PERIOD 

Without  any  complication  as  eegards  the  life  or  duration 
OF  continuing  utility  of  the  asset  created  out  of  the 
LOAN  [continued). 

Loan  borrowed  over  several  years,  in  one  sum  in  each 
year,  repayable  in  one  sum  on  a  certain  specified  date  : 

1.  Where  the  date  of  repayment  is  known  at  the 

time  the  money  is  borrowed. 

2.  Where  the  date  of  repayment  is  fixed  after  the 

sinking  fund  has  been  in  operation  for  a  number 
of  years,  and  an  adjustment  of  the  fund  is 
required. 


Summary  of  the  methods  of  adjustment. 

(1)  Summary  of  the  method  of  ascertaining,  at  the  end  of 
tlie  period  of  construction,  the  future  or  amended  equal  annual 
instalment  to  he  set  aside  and  added  to  the  amount  now  in 
the  fund  which  has  been  provided  by  the  accumulation,  during 
the  jjeriod  of  construction,  of  temporary  instalments  set  aside 
in  respect  of  amounts  borrowed  over  a  series  of  years,  all  of 
which  were,  and  still  are,  repayable  in  one  sum  on  a  certain 
specified  date  which  was  known  at  the  time  the  money  was 
borrowed.  The  problem  is  not  complicated  by  any  variation  in 
the  i^eriod  of  repayment  due  to  the  life  of  the  asset. 

(7)  Ascertain  from  the  actual  records  the  a/nouuf  standing 
to  the  credit  of  the  fund,  at  the  time  the  adpistment  is 
required  to  be  made.  Statement  XXIX  B. 

(2)  Calculate  the  amount  of  loan  which  u-ill  be  provided  at 

the  end  of  the  unexpired  repayment  period^  by  the 
accumulation  of  this  amoit/it  noiv  in  the  fiind^  at  the 
future  rate  of  accumulation. 

Standard  Calculation  Form,  No.  1. 


THE  DATES  OF  BORROWING  AND  REPAYMENT   355 

(3)  Deduct  the  am,ount  so  found ^  as  in  (2),  from  the  amoiint 

of  the  original  loan.   . 

(4)  The  remainder  will  represent  the  balance  of  loan  to  he 

j)rovided  hy  the  accumulation,  at  the  future  rate,  of  the 
required  amended  annual  instalment  to  be  added  to  the 
fund  during  the  tine j^jji red.  repayment  'period. 

{5)  Calculate  the  annual  instalme7it  so  required. 

Standard  Calculation  Form,  No.  3x. 

(6)  The  annual  instahnent  so  ascertained  should  be  equal  to 

the  sum  of  the  several  annual  instalments  already  set 
aside  in  respect  of  the  amounts  borroioed  in  each  year 
provided  there  is  not  any  variation  in  the  period  of 
repayment  or  rate  of  accumulation.  Table  XXIX  A . 

(7)  Any  variation  between  the  annual   instalment  $0  ascer- 

tained, as  in  (J),  and  the  sum  of  tlie  several  annual 
instalments  already  set  aside  will  be  due  to  an  abnormal 
past  accumulation  of  the  fund,  and  will  result  in  a 
surplus  or  deficiency  in  the  amount  of  loan  to  be  provided 
at  the  end  of  the  repayment  period. 

(8)  Such  surplus  or  deficiency  {if  any)  in  the  amount  of  loan 

to  be  ultimately  provided  should  be  corrected  in  the 
manner  already  described  under  these  heads  in  previous 
chapters. 

(.9)  Prepare  a  statement  showing  the  final  repayment  of  the 
loan  by  the  operation  of  the  sinking  fund  under  the 
amended  conditions.  Statement  XXIX.  B. 

{10)  Prepare  the  usual  pro  forma  account. 

(2)  Summary  of  the  method  of  ascertaining,  at  some 
future  time,  the  future  or  amended  equal  annual  instalments 
to  be  set  aside  and  added  to  the  amount  now  m  the  ftmd  which 
has  been  provided  by  the  accumuhition  of  previous  instalments 
set  aside  in  respect  of  loans  borrowed  over  a  series  of  years, 
all  of  wliicli  were  orig^inally  repayable  at  the  end  of  successive 
years,  l)ut  wliicli  are  now  repayable  in  one  sum  on  a  certain 
date  now  for  tlie  first  time  specified.  The  problem  is  not 
complicated  by  any  variation  in  the  period  of  repayment  due 
to  the  life  of  the  asset. 

{1)  Ascertain  from  the  actual  records  the  amount  standing 
to  the  credit  of  the  fund  at  the  time  flic  adjustment  is 
required  to  he  mdde.  Table  XXIX  D. 


356         REPAYMENT   OF    LOCAL   AND    OTHER   LOANvS 

(2)  Calculate  the  amourit  of  loan  which  will  he  lyrovidcd  at 
the  end  of  the  unexinred  repayment  fjeriod  hy  the 
accumuJation  of  this  amount  noiv  in  the  fund  at  the 
future  rate  of  accumulation. 

Standard  Calculation  Form,  No.  1. 

{3)  Deduct  the  amount  so  found,  as  in  (2),  front  the  amount 
of  the  original   loan. 

{4)  The  remainder  ivill  represent  the  balance  of  loan  to  be 
provided  by  the  accumulation  at  the  future  rate  of  the 
required  amended  equal  annual  instalment  to  he  added, 
to  the  fund  during  the  unexpired  repayment  period 
in  substitution  for  the  annual  instalment,  as  originally 
set  aside. 

(5)  Calculate  the  annual  instalment  so  required. 

Standard  Calculation  Form,  No.  3x. 

(6)  Prepare  a  statement  shoiving  the  fnal  repayment  of  the 

loan   by   the  operation   of   the   sinMng   fund   under   the 
amended  conditions . 

(7)  Prepare  the  usual  pro  forma  account. 

The  loans  about  to  be  considered  differ  from  tlie  preceding 
example  only  in  the  fact  tbat,  altbougb  the  borrowings  are 
spread  over  a  series  of  years,  the  loan  is  repayable  in  one  sum 
instead  of  at  the  end  of  successive  years  corresponding  to  the 
number  of  years  during  which  the  money  was  borrowed.  The 
enquiry  is  still  limited  to  loans  in  respect  of  outlay  having  a 
uniform  period  of  repayment.  The  date  of  repayment  of  the 
loan  is  generally  prescribed  in  the  original  sanction  or 
authorisation,  and  may  be  either  (1)  a  specified  date,  (2)  a 
definite  number  of  years  from  the  date  of  the  sanction,  or  from 
the  commencement  of  operations,  or  (3)  a  given  number  of  years 
from  a  date  later  than  the  sanction ;  or,  in  other  words,  a 
deferred  sinking  fund.  On  the  other  hand,  the  date  of  repay- 
ment may  be  fixed  by  the  local  authority  or  by  Parliament 
some  years  after  the  loan  has  been  borrowed  and  a  sinking  fund 
or  funds  established.  This  may  arise  on  the  consolidation  of 
existing  loans,  and  also  under  the  folloM-ing  or  other  similar 
conditions.  A  local  authority  has  obtained  powers  to  construct 
certain  works  and  to  borrow  on  loan,  and  the  power  or  sanction 
provides  that  the  loan  shall  be  repaid  in  25  years  from  the 
dates  of  borrowing.  The  actual  construction  of  the  works 
extends  over  a  period  of  three  years,  and  such  an  amount  only 


THE   DATEvS    OF   BORROWING   AND    REPAYMENT       357 

is  borrowed  iu  each  year  avS  will  pay  for  the  works  actually 
constructed  in  that  year.  At  the  end  of  three  years  the  works 
authorised  are  completed  and  the  full  amount  of  the  loan  has 
been  borrowed.  During  the  period  of  construction  the  proper 
instalments  have  been  regularly  set  aside  out  of  revenvie  or 
rate,  to  provide  the  amount  of  loan  repayable  at  the  end  of  each 
of  the  prescribed  periods  of  25  years.  The  local  authority  then 
decide  to  convert  the  loans  into  stock  redeemable  on  a  fixed 
date. 

This  date  may  be  specified  under  further  powers  granted,  or 
may  be  fixed  by  the  local  authority  at  the  time  of  making  the 
adjustment. 

The  above  instances  may  be  divided  into  two  classes  requir- 
ing different  treatment,  although  the  loan  relates  to  outlay  of 
one  character  only,  namely  :  — 

Class  1.  Loans  in  respect  of  which  the  date  of  repayment  is 
known  at  the  time  the  money  is  borrowed,   and 

Class  2.  Loans  in  respect  of  which  the  date  of  repayment  is 
fixed  after  the  sinking  fund  has  been  in  operation 
for  some  years,  and  an  adjustment  becomes 
necessary. 

The  method  of  making  the  adjustment,  however,  rather  than 
the  cause  of  the  adjustment,  is  the  principal  object  of  enquiry. 

Class  I.  Loans  in  respect  of  one  class  of  outlay  only,  borrowed 
over  a  series  of  years,  repayable  in  one  sum  on  a  specified 
date,  which  date  is  known  at  the  time  the  money  is 
borrowed. 

The  first  example  will  relate  to  a  loan  raised  by  the  issue  of 
stock  repayable  on  a  specified  date.  The  actual  borrowing  is 
spread  over  three  years  (the  period  of  construction  of  the 
works),  and  the  period  of  repayment  is  25  years  from  the 
commencement  of  operations.  It  will  be  assumed  for  the 
purpose  of  simplifying  the  conditions,  that  the  local  authority 
has  borroM-ed  the  money  immediately  prior  to  the  beginning 
of  the  financial  year  and  that  work  has  been  commenced  on 
that  date.  Subsequent  borrowings  are  made  on  the  first  day 
of  the  two  following  financial  years,  and  there  is  not  therefore 
any  complication  due  to  the  loan  being  borrowed  at  various 
dates  in  any  one  year. 

The  rate  of  accumulation  of  the  sinking  fund  is  3|  per  cent. 


358 


REPAYMENT   OF   LOCAE   AND    OTHER   LOANvS 


Altlioug-li  it  is  requisite  to  keep  only  one  sinking  fund,  separate 
calculations  must  be  made  of  the  annual  instalments  to  be 
charged  to  revenue  or  rate,  and  added  to  the  sinking  fund,  in 
respect  of  each  annual  amount  of  loan  borrowed.  The  complete 
conditions  are  shown  in  the  following  table  :  — 

TABLE  XXIX.  A. 

Loan  of  £11,355,  borrowed  over  three  years,  repayable  in  one 
sum  on  a  specified  date. 

Annual  instalments  calculated  for  varying  periods,  all  to 
mature  on  the  same  date.  Rate  of  accumulation  3|  per 
cent. 


Year  of 
borrowing. 

Redemption 
period. 

Amount 
borrowed 
each  year. 

Annual 

instalment 
on  yearly 
borrowing. 

Annual 

instalment 

at  end  of 

each  year. 

First. 

25  years 

3,785 

97-176 

97-176 

Second. 

24  years 

3,785 

103-227 

200-403 

Third. 

23  years 

3,785 

109-836 

310-239 

11,355 

310-239 

It  will  be  noticed  that  the  loan  is  borrowed  in  equal  annual 
amounts  during  the  period  of  construction,  and  that  the  annual 
instalments  in  respect  of  the  several  amounts  borrowed  are 
gradually  increased  owing  to  the  reduction  in  the  period  of 
repayment.  The  instalment  to  be  set  aside  at  the  end  of  the 
first  year  is  £97-176  only  and  increases  until  the  end  of  the 
third  year  when  it  attains  the  maximum  of  £310239,  which 
will  be  continued  for  a  further  22  years  when  the  amount  in 
the  sinking  fund  should  be  £11,355,  provided  care  has  been 
taken,  at  the  end  of  each  year,  to  see  that  the  fund  has  accu- 
mulated at  the  proper  rate  in  accordance  with  the  pro  forma 
account  which  should  have  been  prepared. 

In  this  instance  there  is  not  any  decreasing  instalment 
during  the  later  years  of  the  repayment  period  as  was  the  case 
in  the  previous  example,  Table  XXVIII.  B,  seeing  that 
altliougli  the  borrowing  is  spread  over  three  years  the  instal- 
ments are  calculated  on  the  basis  that  the  whole  of  the  loan 
will  matvire  on  the  same  date.  The  final  repayment  of  the  loan 
is  shown  in  the  following  statement:  — 


THE    DATEvS    OF    BORROWING    AND    REPAYMENT       359 

STATEMENT  XXIX.  B. 

Loau  of  £11,355,  burrowed  over  three  years,  repayable  in  one 
sum  on  a  specified  date. 

Showing  the  final  repayment  of  the  loan  by  the  operation  of 
the  sinking  fund  and  the  annual  instalments  shown  in 
Table  XXIX.  A. 


Amount  in  the  fund  :  — 

At  end  of  first  year,  instalment 

At  end  of  second  year  :  — 

Interest,  3|  per  cent.       ...         £3401 
Instalment £200403 


£97176 


£203-804 


At  end  of  third  year  :  — 

Interest,  3^  per  cent.       ...       £10-534 
Instalment £310-239 


£300-980 


£320-773 


Amount  in  the  fund  at  the  end  of  the  third  year    ...     £621-753 

At  the  end  oi  the  25th  year  the  amount  in  the  fund 
will  be  as  follows:  — 

Amount  of  £621-753  for  22  years  at  3^  per  cent. 

per  annum.   Standard  Calculation  Form,  No.  1       £1325-3 

Amount  of  £310239  per  annum  for  22  years  at 
3^  per  cent,  per  annum 

Standard  Calculation  Form,  No.  3     £10029-7 


Total  amount  of  loan  ... 


£113550 


Class  2.  Loans  in  respect  of  one  class  of  outlay  only,  borrowed 
over  a  series  of  years  repayable  in  one  sum  on  a  specified 
date,  such  date  of  repayment  being  fixed  after  the  sinking 
fund  has  been  in  operation  for  a  number  of  years  and  an 
adjustment  is  required. 

The  second  class  of  loans  borrowed  over  a  series  of  years 
will  now  be  considered,  namely,  those  in  which  the  date  of 
repayment  of  the  whole  of  the  loan  is  fixed  after  the  sinking 
fund  has  been  in  operation  for  some  years,  prior  to  which  time 


36o    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

each  year's  borrowiugs  were  repayable  at  the  end  of  successive 
years.  In  such  a  case  it  is  necessary  to  make  an  adjustment 
in  order  to  ascertain  the  future  annual  instalment  to  be  added 
to  the  sinking  fund  during  the  whole  of  the  newly  ascertained 
redemption  period,  in  substitution  for  the  varying  instalments, 
as  shown  in  Table  XXYIII.  B.  This  adjustment  depends  upon 
two  factors,  namely,  the  amount  now  in  the  fund,  and  the 
exact  date  fixed  for  the  redemption  of  the  AA^hole  of  the  loan. 
Seeing  that  the  loan  in  Table  XXA'III.  B.  was  originally 
repayable  over  a  series  of  four  years,  and  is  now  repayable  on 
one  uniform  date,  it  is  advisable  to  adhere  as  closely  as  possible 
to  the  original  conditions  as  to  repayment,  by  expediting  the 
repayment  of  part  of  the  loan  and  delaying  the  repayment  of  an 
equivalent  part.  In  fact  there  is  here  a  mild  form  of  the 
equation  of  the  period  of  repajanent,  and  an  average  eqviation 
of  two  years  will  be  adopted  since  the  present  example  is  chosen 
to  illustrate  the  method  of  making  the  adjustment  rather  than 
to  demonstrate  the  proper  mathematical  method  of  finding  the 
equated  period  of  repayment.  This  will  be  fully  considered  in 
Chapter  XXXII,  where  it  Avill  be  shown  that  the  ordinary 
arithmetical  method  of  finding  the  equated  period  is  incorrect, 
but  not  to  such  an  extent  as  to  make  any  appreciable  difference 
in  two  years  seeing  that  in  such  calculations  the  nearest 
whole  number  of  years  is  adopted. 

In  the  following  example  the  original  conditions  as  to  the 
amounts  of  loan  borrowed,  and  the  annual  instalments  required, 
are  the  same  as  in  Chapter  XXYIII,  the  only  difference  being 
that  the  loan  is  repayable  in  one  sum  instead  of  at  the  end  of 
four  successive  years.  The  following  table  contains  the  original 
conditions  in  the  example  now"  under  consideration,  and,  as 
regards  the  actual  figures,  is  a  copy  of  Table  XXYIII.  A. 

TABLE  XXIX.  C. 

Loan  of  £26,495,  borrowed  over  four  j^ears,  repayable  in  one 
sum  on  a  specified  date  fixed  after  the  fund  has  been  in 
operation  for  a  number  of  years  and  an  adjustment  is 
required. 

Original  anniuil  instalments  all  calculated  to  mature  in  25 
years,  but  at  the  end  of  successive  years.  Rate  of  accu- 
mulation 3^  per  cent. 

This  table  is  a  copy  of  Table  XXYIII  A. 


THE   DATES   OF   BORROWING    AND   REPAYMENT       361 


Year  of 
borrowing. 

Redemption 
period. 

Proportion 
borrowed 
each  year. 

Amount 
borrowed 
each  year. 

Annual 
instalment 
on  yearly 
borrowing. 

Annual 
instalment 

at  end  of 
each  year. 

First. 

25  years 

^/>4 

3785- 

97-176 

97-176 

Second . 

25  years 

'U. 

5677-5 

145-764 

242-940 

Third. 

25  years 

Vl4 

7570- 

194-353 

437-293 

Fourth. 

25  years 

'U. 

9462-5 
26495 

242-941 

680-234 

680-234 

Tables  XXYIII.  A,  and  XXVIII.  B  show  the  annual  in- 
stalments to  be  set  aside  to  repay  the  above  loans  at  the  end 
of  the  25th,  26th,  27th  and  28th  years. 

As  stated  in  the  preliminary  remarks  in  this  chapter,  during 
the  5th  year  circumstances  arise  which  render  it  necessary  to 
provide  for  the  repayment  of  the  whole  of  the  loan  on  one  date, 
instead  of  at  the  end  of  4  successive  years,  and  it  will  be 
assumed  that  the  end  of  the  26th  year  is  adopted  as  the 
redemption  date.  Four  separate  sinking  funds  have  been  kept 
and  each  fund  stands  at  the  proper  amount  shown  by  the  pro 
forma  account.  This  means  that  the  accumulation  of  each 
fund  by  way  of  income  from  investments  has  been  equal  to  the 
calculated  amount,  or  that  any  deficiency  has  been  made  good 
year  by  year  out  of  revenue  or  rate. 

In  case  there  is  a  deficiency  or  a  surplus  in  the  fund  at  the 
time  of  making  the  adjustment  it  may  be  accurately  adjusted 
if  necessary  by  the  methods  fully  described  in  previous  chap- 
ters, but  as  a  general  rule  unless  the  discrepancy  is  of  large 
amount  it  is  merged  in  the  general  adjustment  about  to  be 
made.  In  ordinary  practice  of  course  the  present  position  of 
the  fund  is  ascertained  from  the  actual  records  or  books  of 
account,  but  in  the  present  example  the  amount  in  the  fund 
must  be  found  by  actual  calculation. 

The  first  step  therefore  is  to  ascertain  the  amounts  which 
should  stand  to  the  credit  of  each  of  the  individual  sinking 
funds  relating  to  each  of  the  four  year's  borrowings  at  the  end 
of  the  fourth  year,  this  being  the  date  when  the  maximum 
instalment  has  been  set  aside  in  respect  of  the  full  amount  of 
the  loan  which  has  then  been  borrowed. 

This  may  be  done  by  the  following  arithmetical  calculation 
which  is  somewhat  shorter  than  by  the  tables  and  logarithms 
and  which  has  the  further  advantage  that  it  shows,  although 
in  decimal  form,  the  actual  entries  in  the  ledger. 


362 


REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 


TABLE  XXIX.  D. 
Loan  of  £26,495,  borrowed  over  four  years,  repayable  in  one 

sum  on  a  specified  date  fixed  after  tlie  fund  lias  been  in 

operation   for  a  number  of  years,   and   an   adjustment    is 

required. 
Separate   sinking   funds.     The    amount    in    each    fund    at    the 

end  of  the  fourth  year,  will  be  as  follows  :  — 


Amount  set  aside 

at  the  end  of  the 

following 

financial  years. 

Sinking  Funds  in  respect  of  loan  borrowed  at 
of  the  following  financial  years. 

First.                 Second.                   Third. 

the  beginning 
Fourth. 

First 

Instalment 

97T76 

Second 

Interest 

:J-4U1 

Instalment 

97176 

145-764 

197-753 

145-764 

Third 

Interest 

6-921 

5-102 

Instalment 

97-176 

145-764 

194-353 

301-850 

296-630 

194-353 

Fourth 

Interest 

10-565 

10-382 

6-802 

Instalment 

97176 

145-764 

194-353 

242-941 

409-591 

452-776 

395-508 

242-941 

395-508 
452-776 
409-591 

Amount  star 

iding  to  the 

credit  of  the  four 

sinking 

funds  at  the 

end  of  the 

fourth 

year  of 

borrowing 

£1500-816 

The  accuracy  of  the  above  calculation  may  be  proved  by 
assuming  that  only  one  sinking  fund  had  been  kept.  Although 
this  is  not  recommended  in  the  case  of  a  loan  repayable  over  a 
series  of  years,  there  is  an  advantage  in  keeping  only  one  fund 
where  the  loan  is  repayable  in  one  sum  provided  a  pro  forma 
account  of  the  operation  of  the  fund  is  prepared  when  the  full 
amount  of  loan  has  been  borrowed.  The  following  table  shows 
the  amount  in  the  sinking  fund  at  the  end  of  the  fourth  year : 

TABLE  XXIX.  E. 

Loan  of  £26,495,  as  above. 

One  sinking  fund  only.     The  amount  in  the  iuiid  at  the  end 
of  the  fourth  year,  will  be  as  follows  :  — - 


THE    DATEvS    OF   BORROWING    AND    REPAYMENT       363 


Vniount  added  to  the 
fund  at  the  end  of 
the  following  years. 

First 

Amount  to  credit 

of  fund  at 
beginning  of  year. 

Annual  accretions. 
Interest  :U  %         Instalment. 

97176 

Amount  to  credit 
of  fund  at 
end  of  year. 

97176 

Second     ... 

97176 

a-401 

242-940 

343-517 

Tliird        ... 

U-S-bll 

12023 

437-293 

792-833 

Fourth     . . . 

792-833 

27-749 

680-234 

1500-816 

43173 

1457-643 

1500-816 

At  the  time  of  luakiiig  the  adjustment  four  sinking  funds 
are  in  operation,  all  relating  to  the  repayment  of  a  loan  of 
£26,495  borrowed  in  unequal  amounts  over  a  period  of  four 
years,  and  each  year's  borrowings  are  repayable  in  25  years 
from  the  date  of  the  original  borrowing,  the  last  portion  of  the 
loan  being  repayable  at  the  end  of  the  28th  year.  The  actual 
repayment  of  the  total  loan  was  originally  spread  over  a  period 
of  4  years,  but  under  the  new  conditions  it  is  required  to  amend 
the  annual  instalment  of  £680234  to  be  set  aside  for  21  years 
followed  by  decreasing  instalments  for  3  years  as  shown  in 
Table  XXVIII.  B,  In  place  of  these  varying  instalments 
required  to  repay  the  loan  at  the  end  of  four  successive  years 
it  is  necessary  to  ascertain  the  annual  instalment  which  will 
repay  the  whole  of  the  loan  of  £26,495  at  the  end  of  22  years 
from  the  present  time,  bearing  in  mind  that  there  is  in  the 
fund  an  amount  of  £1500816  which  can  be  applied  in  reduc- 
tion of  the  future  annual  instalment. 

The    amended    annual    sinking    fund    instalment    may    be 
found  in  the  following  manner  which  is  similar  in  principle  to 
the  annual  increment   (balance  of  loan)   method  described   in 
Chapter  XXII:  — 
Amount  of  loan,  repayable  in  22  years  from  the 

present  time         £26495-00 

Deduct  therefroin  the  amount  of  loan  which  will  be 

provided  by  the  accumulation  at  3^  per  cent. 

for  22  years  of  the  £1500-816  now  in  the  fund. 

By  standard  calculation  form  No.  1 £3199-07 

leavino:  a  balance  of  loan  of     £23295-93 

to  be  provided  by  the  accumulation  at  3^  per  cent,  of  the  future 
amended  annual  sinking  fund  instalment  to  be  set  aside  for 
22  years. 

This    amended    annual    instalment    as    may    be    found    by 
standard  calculation  form  No.  3  x,  is  £720-59. 


364         REPAYMENT   OF    LOCAL   AND    OTHER   LOANvS 

Proof  of  the  above  Adjustment.  In  ordinary  practice, 
of  course,  tlie  best  method  of  proving  the  above  calculation  is 
to  prepare  the  usual  pro  forma  account  so  often  recommended, 
showing  the  amount  which  should  be  in  the  fund  at  the  end  of 
each  year,  and  which  is  required  in  order  to  control  the 
subsequent  accumulation  of  the  fund.  This  method,  however, 
is  unsuited  to  a  work  of  this  nature,  and  it  is  preferable  to 
adopt  a  method  of  proof  based  upon  actuarial  principles. 

To  recapitulate  the  data.  A  loan  of  £26,495  is  repayable 
at  the  end  of  22  years,  and  there  is  in  the  sinking  fund  the  sum 
of  <£1500"816  which  will  accumulate  at  3^  per  cent.  The 
problem  is  to  ascertain  the  sinking  fund  instalment  to  be  set 
aside  and  accumulated  for  the  remaining  22  years. 

The  calculation  is  made  in  two  stages  as  follows  :  — First 
ascertain  the  annual  instalment  to  be  set  aside  and  accumulated 
as  a  sinking  fund  at  3^  per  cent,  to  provide  £26,495  at  the  end 
of  22  years.     This  annual  instalment  is  c£819'54. 

The  next  step  is  to  ascertain  the  annual  sum  or  annuity  by 
which  this  instalment  will  be  reduced  by  the  amount  of 
£1500'816  now  in  the  fund,  or  in  other  words  the  annuity  for 
22  years  at  3^  per  cent,  which  may  be  purchased  by  the  above 
sum  of  £1500' 816.  This  annual  amount,  using  the  author's 
standard  calculation  form  No.  5,  will  be  found  to  be  £9895. 

The  adjusted  annual  instalment  therefore  is:  — 

Annual  instalment  to  repay  the  loan  of  £26,495  in 

22  years     £819-54 

less    the    reduction    therein    due    to    the    amount    of 

£1500816  now  in  the  fund £98-95 


Amended  annual  instalment  as  previously  ascertained     £720-59 


In  the  foregoing  example  it  has  been  assumed  that  the 
amount  of  the  loan  remains  unchanged  and  that  the  rate  of 
accumulation  of  the  fund  and  the  income  from  investments 
will  continue  to  be  3^  per  cent,  as  in  the  original  example  in 
Chapter  XV.  The  only  variation  is  in  the  period  of  repay- 
ment. Tlie  methods  described  in  Cluipter  XXIY,  variation  in 
the  period  of  repayment,  cannot  be  adopted  because  tlie  amended 
annual  instalment  here  required  is  to  replace  four  instalments 
to  l)e  set  aside  for  varying  ])eriods  instead  of  one  instalment 
for  one  period.  The  method  will  apply  equally  to  loans  not 
raised  by  tlie  issue  of  stock  if  the  whole  of  the  loans  are 
repayable  in  one  sum  on  a  specified  date. 


THE    DATE    OF    BORROWING  365 


CHAPTER  XXX. 

SINKING  FUND  PROBLEMS,  RELATING  TO  THE 
DATE  OF  BORROWING  AND  THE  REDEMPTION 
PERIOD, 

Without  any  complication  as  eegaeds  the  life  or  duration 
OF  continuing  utility  of  the  asset  created  out  of  the 
LOAN  [continued). 

Loan  borrowed  in  one  or  more  years  in  varying  amounts  at 
various  dates  in  each  year,  and  it  is  required  that 
the  revenue  or  rate  account  of  each  year  shall  be 
charged  with  a  proportionate  part  of  the  annual 
sinking  fund  instalment. 


The  actual  borrowings  in  any  one  year  (whether  in  respect 
of  a  loan  borrowed  entirely  in  one  year,  or  borrowed  over  a 
series  of  years  depending  upon  the  period  of  construction)  are 
often  made  at  various  dates  during  the  year  because  the  money 
is  not  required  or  is  not  readily  obtainable.  To  carry  out  the 
strict  letter  of  the  obligation  to  repay  the  loan  at  the  end  of  a 
prescribed  number  of  years  from  the  date  of  borrowing  would 
be  practically  impossible  if  each  individual  borrowing  had  to 
be  treated  separately.  The  general  practice  is  to  treat  all  the 
sums  received  in  any  one  year  as  if  they  had  been  borrowed 
at  the  end  of  the  financial  year  and  not  to  set  aside  any  sinking 
fund  instalment  in  respect  of  the  broken  period  of  the  year  of 
borrowing,  the  provision  of  the  first  full  annual  instalment 
being  deferred  until  the  end  of  the  succeeding  financial  year, 
which  simplifies  the  working  of  the  fund  very  considerably. 
In  the  case  of  a  small  loan  borrowed  piecemeal  in  this  fashion 
in  one  year  there  is  not  any  great  objection  to  outweigh  the 
manifest  advantages;  and  the  same  applies  to  loans  borrowed 
over  a  period  of  vears  during  construction  in  which  the  annual 
amount  borrowed  is  not  large.  The  principle  of  deferring  the 
first  annual  contribution  has  been  extended  by  Parliament  in 
certain  cases,  where  the  operation  of  the  sinking  fund  has  been 
suspended  for  a  specified  niimber  of  years. 


366         REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 

But  it  may  happen  in  the  case  of  a  local  authority  or  a 
commercial  or  financial  vindertaking  that  the  loan  is  of  large 
amount  and  may  be  borrowed  during  one  year.  It  may  be 
necessary  and  equitable  in  such  a  case  to  charge  the  revenue 
or  rate  accoimt  of  that  year  with  the  proper  calculated  propor- 
tion of  one  year's  annual  instalment  in  respect  of  each  separate 
borrowing,  based  upon  the  part  of  one  year  for  which  the  under- 
taking has  had  the  use  of  the  money  raised  during  the  year, 
and  not  defer  the  first  annual  contribution  until  the  end  of  the 
succeeding  year. 

Such  an  instance  might  arise  in  connection  with  the  pur- 
chase of  an  existing  undertaking  by  a  local  authority  where 
the  purchase  money  is  payable  by  instalments  spread  over  a 
year  and  is  borrowed  as  and  when  required,  but  the  local 
authority  enters  into  possession  immediately  and  takes  the 
whole  of  the  profits.  If  no  contribution  to  the  sinking  fund 
were  made  during  the  first  year  the  revenue  or  rate  account  of 
that  year  would  show  a  fictitious  profit  as  compared  with 
subsequent  years.  In  such  a  case  it  would  appear  not  only 
equitable  but  good  accounting  jDractice  to  charge  the  revenue 
or  rate  account  of  that  year  with  a  portion  of  the  annual 
instalment  commensurate  to  the  amount  of  loan  it  has  had  the 
use  of  during  part  of  the  year. 

In  the  case  of  a  commercial  undertaking  the  revenue 
account  for  the  year  would  of  course  l>e  charged  with  interest 
upon  the  loan  for  the  exact  number  of  days  the  money  had 
been  borrowed,  and  the  same  would  apply  to  the  revenue 
account  of  a  local  authority  where  the  accounts  are  kept  upon 
the  "  income  and  expenditure "  as  distinguished  from  the 
"  receipts  and  payments  "  system. 

If  the  principle  applies  to  interest  upon  the  loan,  it  should 
certainly  apply  to  the  annual  contribution  to  the  sinking  fund, 
especially  in  the  case  of  a  local  authority  where  both  amounts 
are  specific  charges  against  revenue  or  rate.  In  the  case  of  a 
commercial  undertaking  the  conditions  as  to  a  sinking  fund  are 
much  more  elastic  than  is  the  case  with  the  loans  of  local 
authorities,  and  mucli  would  depend  upon  the  actual  conditions 
laid  down  in  tlie  deed  governing  the  loan,  which  would  be  taken 
into  account  by  the  auditors  before  certifying  the  accounts. 

In  the  case  of  local  authorities  it  is  impossible  to  lay  down 
any  hard  and  fast  rule.  The  conditions  imposed  upon  such 
authorities  have  of  late  years  been  of  a  uniform  nature  depend- 
ing upon  the  probable  life  of  the  asset,  but  where  powers  are 
granted  by  special  Act  of  Parliament  wider  latitude  has  often 


THE    DATE    OF    BORROWING  367 

been  allowed,  and  tlie  special  nature  of  the  powers  requires 
careful  scrutiny  in  eacli  case.  Attention  may,  however, 
properly  be  directed  to  the  magnitude  of  the  loan;  in  some 
instances  the  amount  involved  may  be  considerable,  and  may 
point  to  the  necessity  of  making  some  such  adjustment,  but  to 
insist  upon  it  in  all  cases,  irrespective  of  the  amount  of  the  loan, 
might,  and  possibly  would,  involve  considerable  labour  without 
any  corresponding  advantage. 

With  regard  to  the   actual   adjustment,    there   are  several 
interesting  points,  and  the  problem  is  not  so  simple  as  it  appears 
at  first  sight.     To  find   the   actual   proportion   of  the   annual 
instalment  to  be  charged  to  the  revenue  account  of  the  year  of 
borrowing  it  is  first  necessary  to  ascertain  the  annual  instalment 
to  repay  the  total  loan  borrowed,  having  regard  to  the  redemp- 
tion period  imposed.     Seeing  that  an  adjustment  of  this  nature 
is  rarely  made  in  the  case  of  small  loans,   but  is  confined  to 
loans  of  considerable  magnitude,  it  is  very  important  that  the 
calculation  should  be  made  with  extreme  accuracy.     Such  large 
loans  are  generally  raised  by  the  issue  of  stock  redeemable  on  a 
fixed  date,  and  it  often  happens  that  the  total  amount  of  the 
loan  is  borrowed  over  a  period  of  years,  rendering  it  necessary 
to  make  a  similar  calculation  of  the  proportionate  part  of  one 
year's  annual  instalment  at  the  end  of  each  year  of  borrowing. 
In  this  manner  varying  amounts  are  added  to  the  fund  each 
year,  which  departs  from  the  normal  growth  of  a  sinking  fund 
by  equal   annual   instalments.     This  will  render  it  necessary 
to  set  aside  each  year  what  may  be  termed  temporary  instal- 
ments, and  to  adjust  the  fund  when  the  whole  of  the  loan  has 
been  raised,  by  ascertaining  the  exact  equal  annual  instalment 
required  to  be  set   aside   during   the   remaining  years   of  the 
redemption  period  to  repay  the  loan   on  the  prescribed   date, 
having  regard  to  the  amount  in  the  fund  at  the  time  of  making 
the  adjustment. 

The  problem  will  be  illustrated  by  a  sinking  fund  to  repay 
a  loan  of  £11,355  in  one  sum,  on  a  specified  date  (namely,  at 
the  end  of  25  years)  with  a  rate  of  accumulation  of  3|  per  cent., 
the  loan  being  borrowed  in  three  equal  annual  sums  of  £3,785. 
These  amounts  are  borrowed  at  various  dates  during  the  several 
financial  years,  and  it  is  required  that  the  revenue  or  rate 
account  of  each  year  shall  be  charged  with  a  proportionate  part 
of  the  sinking  fund  instalment  in  respect  of  the  money  borrowed 
during  the  year. 

A  similar  loan  has  already  been  used  to  illustrate  the 
example   in    Chapter    XXIX,    in    which    case   the   money   was 


368         REPAYMENT    OF    LOCAL   AND    OTHER   LOANS 

supposed  to  be  borrowed  on  the  first  day  of  each  financial  year, 
and  the  conditions  shown  in  Table  XXIX.  A.  will  be  adopted 
in  the  present  instance  in  order  to  show  the  effect  as  compared 
with  that  example,  although  the  amounts  are  small.  But  the 
principle  is  the  same,  and  the  effect  upon  a  larger  loan  will  be 
readily  appreciated  when  it  is  remembered  that,  given  the  same 
number  of  years  and  rate  of  accumulation,  the  annual  instal- 
ment is  always  proportionate  to  the  loan. 

A  further  imaginary  factor  must  be  assumed,  namely  the 
precise  date  or  dates  in  each  year  on  which  the  loan  was  raised. 

It  is  more  than  probable  that  the  loan  borrowed  in  any  one 
year  will  be  raised  in  more  than  one  sum.  In  such  cases  it  is 
sufficiently  if  not  quite  correct  to  proceed  by  the  arithmetical 
method  and  multiply  the  several  amounts  borrowed  by  the 
number  of  days  between  the  respective  dates  of  borrowing  and 
the  end  of  the  financial  year.  The  sum  of  these  products 
divided  by  the  total  amount  borrowed  during  the  year  and  the 
result  again  divided  by  365  is  the  proportion  of  the  year 
required. 

The  following  example  will  make  the  matter  clear :  — 

TABLE  XXX  A. 

To  ascertain  the  proportion  of  the  annual  instalment  in  respect 
of  the  amounts  borrowed  during  one  year. 

The  Arithmetical  Method. 


Number  of  days 

Product 

Amount 

to  end  of 

of  amount 

borrowed. 

financial  year. 

X  days. 

Date  of  borrowing. 

January  31  £100  334  33,400 

March  31  £200  245  49,000 

June  30  £300  184  55,200 

September  30  £400  92  36,800 


Total     £1000  174,400 


The  eqiiivalent  proportion  of  one  year  for  which  the  under- 
taking lias  liad  the  benefit  of  the  £1000  is  arrived  at  as 
follows  :  — 

174,400  174-4 

^-365  = 

1000  365 

and  this  proportion  of  the  annual  sinking  fund  instalment  is 
chargeable  against  the  revenue  or  rate  account  of  the  year  of 
borrowing. 

In  order,  however,  to  vsimplify  the  following  calculation  it 


THE    DATE    OF    BORROWING  369 

will  be  assumed  that  the  loan  was  raised  in  each  year  in  one 
sum,  and  that  the  local  authority  had  the  use  of  the  money  for 
the  following  portions  of  each  year  :  — 

First  year one  half  of  the  year 

Second  year       one  third  of  the  year. 

Third  year one  quarter  of  the  year. 

The  exact  dates  of  borrowing  during  each  year  have  a  very 
important  effect  upon  the  variation  in  the  annual  instalment 
during  the  period  of  borrowing  and  the  subsequent  period  of 
repayment.  If  the  amounts  are  borrowed  during  the  early 
part  of  the  year,  the  proportionate  part  of  one  year's  instalment 
will  be  greater  than  if  the  money  were  borrowed  during  the 
later  part  of  the  year. 

The  complete  conditions  are  shown  in  the  following  table  :  — 

TABLE  XXX  B. 

Loan  of  £11,-355  borrowed  over  3  years,  repayable  in  one  sum 
on  a  specified  date,  by  means  of  an  annual  sinking  fund 
instalment  to  accumulate  at  3|  per  cent.  The  revenue  or 
rate  account  of  each  year  to  be  charged  with  a  propor- 
tionate part  of  the  annual  instalment  in  respect  of  the 
amount  of  loan  borrowed  during  such  year. 


Annual  amounts  borrowed  and  yearly  and  proportionate  instal- 
ments . 


Annual  instalment. 
Poi'tion  of 
year  for  Period  Proportionate 


Amount     wliicli  money  in  wliicli  part  of  first 

Year.                    borrowed,     borrowed.  repayable.  Yearly.  year's  instalment. 

First             3785           i  25  years  97-176         48-588 

Second         3785           I  24  years  103-227         34409 

Third           3785           i  23  years  109-836         27-459 


£11,355  £310-239 

]SfoTE. — This  table  should  be  compared  with  Table  XXIX.  A. 

The  above  annual  instalments  are  calculated  for  even  periods 
of  25,  24  and  23  years  respectively,  and  in  the  following 
example  it  will  be  assumed  that  they  are  set  aside  during  the 
three  years,  at  the  end  of  Avhich  period  the  necessary  adjustment 
will  be  made.  This  is  the  most  practical  way  of  dealing  with 
the  matter,  although  it  may  properly  be  contended  that  the 
above  yearly  instalments  should  be  slightly  reduced  in  conse- 
quence of  the  proportionate  parts  set  aside  in  respect  of  the  year 
of  borrowing.     The  main  object  of  the  adjustment  is  to  ensure 


370         REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 

that  the  revenue  or  rate  accoimt  of  the  year  of  borrowing 
shall  be  charged  with  a  proper  proportion  of  the  sinking  fund 
instalment  rather  than  that  subsequent  years  shall  be  charged 
to  a  fraction  with  the  exact  mathematical  amount. 

The  following  table  shows  the  complete  and  proportional 
instalments  which  will  be  added  to  the  fund  during  the  three 
years,  and  the  amount  which  should  be  in  the  fund  at  the  end 
of  the  third  year  of  borrowing.  The  broken  year  during  which 
the  first  amount  was  borrowed  is  not  included  in  the  period  of 
repayment,  which  is  in  effect  extended  by  part  of  a  year. 

STATEMENT  XXX  C. 

Loan  of  £11,355,  borrowed  over  three  years,  repayable  in  one 
sum  on  a  specified  date. 

A  proportion  of  each  annual  instalment  to  be  set  aside  in  respect 
of  the  amounts  borrowed  during  each  year. 

The  amount  in  the  sinking  fund  at  the  end  of  each  year  of 
borrowing,  will  be  as  follows  :  — 

Borrowing  begins 
at  beginning  of  fi.rst  year  of  fund  :  — 

I  of  £97-176,  instalment,  first  year...  48588 

Eepayment  period  begins 

at  end  of  first  year  of  fund :  — 

Interest  on  £48-588 1700 

Instalment,  first  year 97-176 

i  of  £10-3-227,  instalment,  second  vear         34-409 

133-285 


181-873 


at  end  of  second  year  of  fund  :  — 

Interest  on  £181-873 6306 

Instalment,  first  year 97-176 

Instalment,   second  year 103-227 

i  of  £109-836,  instalment,  third  year  27-459 


234-228 
416101 


THE    DATE    OF    BORROWING  371 


Borrowing  ceases 


at  end  of  third  year  of  fund:  — 

Interest  on  £416101 14'564 

Instalment,  first  year 9T"176 

Instalment,   second  year 103"227 

Instalment,  third  year      109-836 


324-803 


Amount  in  the  fnnd  at  the  end  of  the  third  year  ...     £740-904 

The    above    amounts    credited    to    the    sinking    fund     are 
contributed  as  follows  :  — 

Charged  to        Interest 
revenue  from 

account.      investments.  Total. 

First  year  of  borrowing 48-588         —  48-588 

First  year  of  repayment  period  131-585       1-700  133-285 

Second  year  of  repayment  period  227-862       6-366  234-228 

Third  year  of  repayment  period  310-239  14564  324803 

718-274     22-630     740904 


as     compared     with     the     previous 
example  in  Statement  XXIX  B.  : 

First  year  of  repayment  period       97-176  —  97176 

Second  year  of  repaVment  period     200403  3-401  203-804 

Third  year  of  repayment  period     310239  10534  320-773 

607-818  13-935  621-753 


or  a  surplus  of 110-456       8695     119151 


There  is  in  the  fund  at  the  end  of  the  third  year 

the  sum  of     "    ...     £740-904 

as  compared  with  the  previous  example, 

Statement  XXIX.  B.     £621-753 


a  surplus  of    £119-151 

being  the  accumulation  of  the  proportionate  parts  of  the  instal- 
ments set  aside  in  respect  of  the  years  of  borrowing,  as  may  be 
verified  bv  a  similar  calculation. 


372    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Seeing  that  the  kjau  is  the  same  in  amount  and  the 
unexpired  period  is  22  years  in  each  case  this  surplus  will  tend 
to  reduce  the  annual  instalment  of  £310'239. 

The  reduced  annual  instalment  may  be  found  in  the  follow- 
ing manner  which  is  similar  in  principle  to  the  annual  incre- 
ment (balance  of  loan)  method  described  in  Chapter  XXII :  — 

Amount   of   loan   repayable    in   22   years    from   the 

present  time    £llo5500 

Deduct  therefrom   the   amount   of   loan   which   will 

be  provided  by  the  accumulation  at  3^  per  cent. 

for  22  years  of  the  £740904  now  in  the  fund. 

By  standard  calculation  form,  No.  1       £15T9'24 

leaving  a  balance  of  loan  of       £9T75'T6 

to  be  provided  by  the  accumulation,  at  31  per  cent.,  of  the 
future  amended  annual  sinking  fund  instalment  to  be  set  aside 
for  22  years. 

This  amended  annual  instalment,  as  may  be  found 

by  standard  calculation  form,  No.  3x,  is £302'38 


Peoof  of  the  above  Adjustment.  The  accuracy  of  the 
above  adjustment  may  be  proved  in  a  similar  manner  to  that 
adopted  in  Chapter  XXIX.  A  loan  of  £11,355  is  repayable 
at  the  end  of  22  years,  towards  which  there  is  in  the  fund  an 
amount  of  £740-904,  which  will  accumulate  at  3^  per  cent. 

The  annual  instalment  to  repay  the  loan  of  £11,355 
in  22  years  at  3^  per  cent.,  as  may  be  found  by 
standard  calculation  form,  No.  3x,  is       £351"~.3 

but  the  amount  of  £740-904  now  in  the  fund  is 
equivalent  to  an  annual  instalment  for  the 
same  period,  as  may  be  found  by  standard 
calculation  form,  No.  5,  of £48"85 


leaving  a  reduced  annual  instalment,  as  previously 

ascertained,   of        £30^"3(S 

Two  methods  have  now  been  described  of  repaying  a  loan 
of  £11,355  (borrowed  over  a  period  of  3  years)  nt  tlie  end  of 
25  years  und(>r  two  sets  of  condiiions,  namely:  — 


THE    DATE    OF    BORROWING 


373 


A,  where  the  annual  instalment  is  set  aside  at  the  end  of 

the  financial  year  following  the  year  of  borrowing,  and 
the  revenue  or  rate  account  of  the  year  of  borrowing 
is  relieved  of  any  charge  in  respect  of  the  sinking  fund 
instalment.  Chapter  XXIX,  Table  XXIX.  B. 

B,  where  the  revenue  or  rate  account  of  each  year  of  borrow- 

ing is  charged  with  a  proportionate  part  of  the  annual 
instalment.  Chapter  XXX,  Table  XXX.  C. 

The  annual  charges  to  revenue  or  rate  account  in  each  case 
may  be  usefully  compared  by  means  of  the  following  table  :  — 


TABLE  XXX.  D. 

Loan  of  £11,355  borrowed  over  three  years,  repayable  in  one 
sum  on  a  specified  date. 


A.  Annual  instalments  only. 


Table  XXIX.  B. 


B.  Annual  and  proportional  instalments 


Table  XXX.  C. 


Comparison  of  the  annual  charges  to  revenue  or  rate  in  respect 
of  the  sinking  fund  instalment. 


Amount  charged  to  the 
revenue  or  rate  account. 


First  year  of  borrowing  ... 
First    year    of    repayment 

period  

Second  year  of  repayment 

period  

Third   year    of   repayment 

period  

Total 

each  of  the  subsequent  22 
years  of  the  repayment 
period  


A. 

Where  the  year  of 

borrowing  is 

relieved  of  any 

charge  in  respect 

of  the 

sinking  fund 

instalment. 

Table  XXIX.  B. 

Xil 


97176 


607-818 


310-239 


Where  the  year  of 

borrowing  is 

charged  with  a 

proportionate  part 

of  the 

sinking  fund 

instalment. 

Table  XXX.  C. 

Excess 

of  B. 

over  A. 

48-588 


131-585 


718-274 


302-380 


48-588 


34-409 


200-403    227-862    27-459 
310-239    310-239     Nil 


110-456 


7-859 


374         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 

The  effect  of  charging  the  revenue  or  rate  account  of  the 
year  of  borrowing  with  a  proportionate  part  of  the  sinking  fund 
instalment  instead  of  deferring  any  charge  to  the  end  of  the 
following   financial   year    may    be    summarised    as    follows :  — 

The  charge  to  revenue  or  rate  account  is  antedated  by  one 
year  to  the  extent  of  the  proportionate  instalment  in  respect  of 
the  first  year's  borrowings,  and  the  same  applies  to  each  year 
during  which  the  borrowing  takes  place.  The  difference 
between  the  two  methods  affects  only  the  revenue  or  rate 
accounts  of  the  years  of  extended  borrowing,  but  as  the  first 
broken  year  of  borrowing  is  not  included  in  the  repayment 
period,  the  annual  instalment  charged  to  revenue  or  rate  in 
the  third  year  of  the  fund  is  the  same  in  both  methods. 

Having  thus  charged  the  earlier  years  with  a  greater  part 
of  the  repayment  burden,  it  is  obvious  that  the  later  years  will 
be  correspondingly  relieved,  and  the  above  table  shows  this 
to  be  the  case.  But  the  increased  burden  to  revenue  or  rate 
account  during  the  years  of  borrowing  is  spread  over  a  smaller 
number  of  years  than  the  relief  is  obtained  during  the  remainder 
of  the  repayment  period,  in  consequence  of  which  the  effect  is 
to  charge  the  revenue  or  rate  accounts  of  the  years  of  borrowing 
with  a  far  greater  annual  amount  than  that  by  which  subse- 
quent years  are  relieved. 

In  the  above  example  the  three  years  of  borrowing  are 
charged  with  an  additional  amount  of  £110-456,  or  an  average 
of  £;j6'818  per  annum,  whereas  the  subsequent  years  are 
relieved  to  the  extent  of  <£7'859  per  annum  only. 

The  above  amounts  of  additional  burden  during  the  earlier 
years  and  the  corresponding  amounts  of  relief  during  the  later 
years  must  not  be  accepted  as  an  exact  ratio  Avhich  will  apply 
to  all  examples  of  this  nature,  because  the  dominant  varying 
factor  in  the  foregoing  adjustment  is  the  actual  date  or  dates 
in  each  year  upon  which  the  loan  was  borrowed.  If  the  loan 
had  been  borrowed  on  the  first  day  of  the  financial  year  the  two 
methods  would  yield  exactly  similar  results,  but  if  the  loan  had 
been  borrowed  during  the  early  part  of  the  previous  year  the 
results  would  have  shown  much  more  variation  than  the  average 
example  used  to  illustrate  the  subject. 

Tlie  necessity  to  make  an  adjustment  of  this  nature  therefore 
depends,  primarily,  upon  the  magnitude  of  the  loan,  and, 
secondly,  upon  the  portion  of  the  year  during  which  the  money 
borroAved  has  been  utilised. 


Section   VI. 

The  Life  or   Duration  of  Continuing  Utility 

of  the  Asset  Created  out  of  the  Loan, 

and  its  Relation  to  the  Redemption  Period 

and  the  Incidence  of  Taxation. 


377 


CHAPTER    XXXI. 

THE  LIFE  OE  DURATION  OF  CONTINUING  UTILITY 
OF  THE  ASSET  CREATED  OUT  OF  THE  LOAN, 
AND  ITS  DELATION  TO  THE  REDEMPTION 
PERIOD  AND  THE  INCIDENCE  OF  TAXATION. 


lu  the  case  of  tlie  loans  of  municipal  or  other  local 
authorities,  there  is  a  further  factor  which  requires  serious 
consideration,  namely,  the  periods  allowed  by  Parliament  (or 
the  Government  Department  concerned)  for  the  repayment  of 
loans  authorised  for  different  classes  of  outlay  having  longer 
or  shorter  lives  or  periods  of  duration  or  utility,  and  this 
variation  in  the  life  of  the  asset  may  in  its  turn  react  upon 
the  period  over  which  the  loaii  is  borrowed  or  is  repayable. 
This  factor  gives  rise  to  the  necessity  to  equate  the  period 
during  which  loans  shall  be  repayable  depending  upon, 

1.  The    life   of   the    asset    and    the    consequent    period    of 

repayment. 

2.  The  date  or  dates  of  borrowing,  whether  in  one  year  or 

spread  over  a  period  of  years. 

3.  A  combination  of  both  periods,   namely,   of  borrowing 

or  repayment. 

This  is  the  most  difficult  problem  in  municipal  finance,  upon 
which  there  is  much  divergence  of  opinion,  as  is  only  natural 
considering  the  extended  and  complicated  nature  of  municipal 
activity,  which,  as  all  who  have  paid  attention  to  such  matters 
know,  is  ever  widening. 

Communities  have  not  any  capital  beyond  the  liability  of 
each  citizen  of  both  the  present  and  future  generations  to 
contribute  his  rateable  proportion  of  the  cost  of  the  benefits 
which  he  receives  from  the  joint  efforts  of  the  community. 
Such  benefits  are  received  by  each  citizen  in  each  generation 
year  by  year,  and  should  be  paid  for  as  and  when  received. 
In  a  primeval  community  individual  benefit  is  paid  for  by 
individual  labour,  but  such  an  ideal  method  of  contribution 
can   only   exist   in   a   small   community,    and  the   difficulty   of 


37S    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

apportioning  the  annual  burden  in  a  rapidly  growing  one  is 
intensified  in  a  far  greater  ratio  than  the  actual  numerical 
increase  of  population. 

Year  by  year,  as  the  community  grows,  the  problem  becomes 
more  complicated.  Works  of  public  utility,  which  in  a  small 
community  might  be  ignored  or  neglected,  become  of  vital 
importance,  and  must  be  carried  out,  and  in  doing  so  regard 
must  be  had  not  only  to  the  present  requirements,  but  also 
to  the  future  growth.  It  is  obviously  useless  to  undertake 
IHiblic  works  which  it  is  well  known  will  be  utterly  inadequate 
to  provide  for  the  needs  of  future  generations,  and  provision 
must  be  made  in  advance. 

This  increases  the  cost  of  all  works  of  public  utility,  and 
involves  the  immediate  spending  of  large  sums  of  money  which 
cannot  be  found  by,  and  cannot  properly  be  charged  against, 
the  present  generation  of  ratepayers,  either  at  once  or  spread 
over  comparatively  iew  years.  Such  outlay  can  only  be  met  by 
pledging  the  credit  of  the  community  for  the  purpose  of  raising 
a  loan.  Consequently  the  repayment  of  the  loan  must  be 
spread  over  an  extended  period  depending  upon — 

(1)  the  probable  life  of  the  asset  upon  which  the  money  is 
expended ; 

(2)  the  liability  of  future  generations  to  provide  further 
works  of  public  utility  which  may  then  be  required;   and 

(3)  the  judgment  of  those  immediately  responsible  for  the 
adequacy  of  the  present  outlay,  including  in  such  term  not  only 
the  actual  permanence  of  the  work  undertaken,  but  also  the 
probability  that  future  advancements  in  knowledge  may  render 
such  works  either  inadequate  in  design  or  too  costly  in  operation. 
This  throws  the  responsibility  of  the  actual  outlay  upon  those 
who  incur  it,  and  it  is  now  a  generally  accepted  principle  that 
the  cost  of  all  outlay  upon  works  of  public  utility  should  be 
written  oif,  and  the  loans  raised  therefor  actually  repaid,  out  of 
current  revenue  or  rate  during  a  period  well  within  the  life 
of  the  particular  works  to  provide  which  the  loan  is  borrowed. 
It  is  obvious,  therefore,  that  the  provision  of  public  utilities 
adequate  to  the  needs  of  future  generations  in  any  individual 
community  is  far  too  great  a  burden  to  be  imposed  upon  the 
present  gem^ration  of  ratepayers,  and  that  this  involves  pledging 
the  future  credit  of  the  community.  By  a  parity  of  reasoning 
the  increase  in  size  and  number  of  communities,  and  the  ever 
widening  sphere  of  local  activities,  renders  it  imperative  that 
the  extent  to  which  the  present  generation  shall  be  allowed  to 


THE    LIFE    OF    THE    ASSET  379 

pledge  the  credit  oi  the  iuture  should  be  treated  not  as  a  local 
but  rather  as  a  national  question.     At  the  present  time,  there- 
fore, all  loans  raised  by  local  authorities  for  purposes  of  public 
utilities  are  subject  to  the  final  approval  of  Parliament,   but 
owing  to  the  enormous  increase  in  this  direction  Parliament  has 
been  compelled  to  delegate  its  powers  as  to  detail  to  Committees 
and  to  certain  Government  departments.     This  has  been  a  very 
gradual    process    extending    over   many  jyears,    during    which 
time  many  Acts  have  been  placed  upon  tlie  Statute  Book,  with 
the  result  that  powers  have  been  obtained  under  both  General 
and  Special  Acts,  and  this  has  led  to  considerable  difference  in 
practice.     The   great   disadvantage   of   this   variation    consists 
in  the  fact  that  the  larger  municipalities,  instead  of  seeking 
powers   under   General   Acts,   may,    in  many   cases,   avoid  the 
careful   scrutiny   of   the  permanent    Government   departments 
(which    now    proceed    upon    regularly    defined    principles)    by 
applying  to  Parliament  for  a  Special  Act.     All  such  Special 
Acts  are  referred  to  Committees  composed  of  members  of  both 
Houses  of  Parliament,  but  there  is  not  any  continuity  in  the 
membership     of    such    Committees,     and    as    the    permanent 
Government  departments  are  not  represented  thereon,  there  is 
not  any  uniformity  of  practice,   and  the  result  is  seen  in  the 
extreme  variation  in  the  powers  as  to  borrowing  and  repayment 
now  existing.     The  present  general  policy  of  Parliament  and 
of  the  Government  departments  charged  with  the  duty  of  fixing 
the  respective  periods  of  repayment  operates  in  the  direction  of 
equalising  the  period  of  repayment  and  the  life  of  the  asset, 
although   the    conditions   now   in    force    vary  considerably    in 
individual  cases  for  the  reasons  already  stated. 

This  principle  is  of  modern  growth.  In  the  early  days  of 
municipal  government,  i.e.,  prior  to  1847,  the  Acts  authorising 
expenditures  upon  public  utilities  did  not  impose  any  obligation 
of  any  kind  to  repay  the  loan  out  of  annual  rates  to  be  levied 
upon  the  community,  and  there  are  to-day  many  loans  out- 
standing in  respect  of  which  no  such  obligation  exists,  and  the 
debt  and  the  interest  payable  thereon  may  for  all  practical 
purposes  be  considered  as  a  perpetual  charge  upon  the  annual 
rates  to  be  levied  by  the  municipalities  unless  and  until  they 
voluntarily  provide  for  its  redemjition  by  making  annual 
charges  against  revenue  or  rate.  In  some  cases  this  provision 
has  been  made  on  the  initiation  of  those  responsible  for  the 
financial  administration  of  the  municipality,  and  in  other  cases 
such  delayed  provision  has  been  imposed  by  Parliament  as  a 
condition  precedent  to  the  granting  of  further  borrowing  powers. 


38o  REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 

There  is  now,  however,  a  considerable  body  of  municipal  opinion 
that  where  the  money  borrowed  is  expended  in  the  purchase 
of  land  in  or  near  the  centre  of  a  city,  and  the  erection  thereon 
of  buildinirs  of  a  substantial  nature  and  of  assured  future 
utility,  the  asset  may  be  considered  as  of  permanent  and  in 
many  cases  even  of  improving  value,  and  that  there  is  not 
therefore  any  necessity  to  burden  the  ratepayers  of  the  present 
or  any  future  generation  with  any  charge  in  respect  of  the 
redemption  of  the  debt  beyond  the  annual  interest  payable  upon 
the  loan  which  interest  may,  it  is  contended,  properly  be 
considered  as  the  equivalent  of  an  annual  rent. 

In  support  of  an  argument  of  this  nature  it  is  contended 
that  local  authorities  may,  and  very  often  do,  occupy  lands  and 
premises  as  ordinary  tenants,  paying  therefor  the  usual  rents 
demanded  by  the  owners  of  the  property,  and  such  tenancies 
may  be  of  an  annual  nature  or  be  by  way  of  lease  for  a  term  of 
years.  Such  leases  for  years  may  be  of  short  duration,  but,  on 
the  other  hand,  they  may,  in  certain  districts,  be  for  very  long 
terms,  possibly  longer  than  would  be  granted  by  Parliament 
for  the  repayment  of  a  loan  authorised  for  the  purchase  of  the 
property.  In  such  cases  it  is  obviously  to  the  advantage  of  the 
local  authority  to  acquire  the  property  by  way  of  lease  rather 
than  by  purchase,  seeing  that  there  will  not  be  any  burden  in 
respect  of  the  sinking  fund  instalment  for  the  redemption  of 
the  loan.  Especially  does  this  apply  to  the  acquisition  of  land 
or  buildings  which  do  not  immediately  require  any  large  outlay 
or  where  the  outlay  is  of  such  a  character  that  it  may  be  spread 
over  a  number  of  years  and  be  met  by  charging  it  direct  to 
current  annual  revenue  or  rate,  or  where  the  annual  outlay  may 
be  so  arranged  that  it  is  less  than  the  sinking  fund  instalment 
to  be  set  aside  to  repay  the  loan  necessary  to  be  raised  to 
purchase  the  property.  Such  conditions  may  not  always  exist, 
especially  in  the  case  of  outlay  in  respect  of  land  required  for 
purposes  of  public  parks  or  open  spaces,  or  large  2)ublic  build- 
ings, such  as  town  halls,  requiring  a  large  expenditure  upon 
buildings,  but  the  principle  is  important  and  may  be  applied 
to  the  occupation  of  land  and  buildings  without  imposing  any 
burden  upon  the  present  and  future  generation  of  ratepayers 
for  the  acquisition  of  properties  which  may  at  any  future  time 
be  replaced  by  others,  which  may  be  not  only  as  cheaply 
acquired  but  may  be  more  suitable  for  the  purpose.  As  against 
this  it  is  argued  that  land  in  the  centre  of  a  city  required 
for  the  erection  of  a  town  liall,  or  land  for  public  parks,  increases 
rapidly  in  value,  and  at  the  end  of  a  long  lease  the  fine  or 


THE    LIFE    OF    THE    ASSET  381 

premium  payable  on  renewal  of  the  lease  would  be  very  large, 
and  tbe  probability  of  such  a  burden  being  laid  by  the  present 
upon  the  slioulders  of  a  future  generation  would  certainly  not 
be  sanctioned  by  Parliament. 

Markets.  The  argument  appears  to  be  equally  strong  when 
applied  to  markets  which  generally  occupy  land  near  the  centre 
of  the  city  and  in  respect  of  which  the  cost  of  the  land  is  the 
predominant  factor,  since  the  buildings  are  not  usually  of  an 
expensive  character.  In  addition  to  the  improving  value  of 
the  site,  markets  are  a  source  of  revenue  consisting  of  tolls 
upon  produce  and  rents  of  floor  space  and  buildings,  which 
revenue,  after  providing  for  all  charges,  yields  a  surplus  which 
is  applied  in  aid  of  the  rates  levied  upon  the  general  body  of 
ratepayers.  In  most  cases  markets  yield  a  surplus  revenue 
over  and  above  all  upkeep  charges,  and  it  seems  only  proper 
that  the  present  generation  of  ratepayers  should  out  of  such 
surplus  revenue  provide  an  annual  instalment  towards  the 
redemption  of  the  debt  before  applying  any  profits  in  aid  of 
their  annual  rateable  contributions  towards  the  upkeep  of  the 
city. 

Water.  The  provision  of  a  permanent  supply  of  pure 
water  for  sanitary  and  other  purposes  is  the  prime  necessity  of 
all  communities  for  many  weighty  reasons,  and  demands  special 
consideration.  The  paramount  factor  in  this  case  is  the 
imperative  obligation  to  provide  for  the  needs  of  the  community 
for  a  number  of  generations  far  in  excess  of  that  requisite  in 
the  case  of  any  other  public  utility;  indeed,  it  may  properly 
be  contended  that  it  is  the  duty  of  the  present  generation  to 
ensure  that  a  permanent  supply  of  pure  water  sufficient  for  the 
needs  of  the  community  shall  continue  for  ever.  Methods  of 
lighting,  transportation,  sewage  disposal  and  other  communal 
necessities  are  being  constantly  improved,  and  any  future 
improvements  in  such  comparatively  minor  utilities  may  be 
carried  out  upon  land  already  allocated  to  them  and  acquired 
by  the  municipality.  But  with  water  supply  the  conditions  are 
the  exact  opposite.  Owing  to  the  rapid  growth  of  cities  involv- 
ing increasing  demands  for  water  for  sanitary  and  maniifactur- 
ing  purposes,  the  natural  areas  suitable  for  the  supply  of  water 
are  being  year  by  vear  continually  encroached  upon  and 
reduced,  and  future  improvements  in  methods  of  transportation 
will  enable  manufarturinc;  processes  to  be  profitably  carried  on 
far  beyond  the  present  city  limits.  Such  conditions  are  favour- 
able to  the  creation  of  vested  interests  in  all  land  which  is  a 


382  REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 

natural  water  area,  and  such  vested  interests  will  be  scattered 
in  such  a  mt^nner  as  to  render  their  acquisition  at  some  future 
time  practically  impossible  even  at  any  jDrice.  It  is  therefore 
the  duty  of  all  municipalities  to  protect  and  preserve  all  natural 
water  areas  for  the  public  use  and  to  expend  money  upon  the 
purchase  far  in  advance  of  present  requirements.  There  is  here 
an  obligation  to  pledge  the  credit  of  the  community  for  the 
purchase  of  land  and  the  construction  of  works  to  provide  a 
sufficient  supply  of  water  to  meet  the  maximum  needs  of  the 
community,  and  yet  the  present  policy  of  Parliament  is  to  allow 
a  shorter  p'eriod  than  formerly.  Owing  to  the  reasons  already 
mentioned  such  land  is  continually  increasing  in  value,  and 
many  existing  water  undertakings  are  now  worth  very  much 
more  than  their  original  cost.  It  is  therefore  argued  that  the 
repayment  of  money  borrowed  to  provide  the  cost  of  land  for 
water  areas  should  be  spread  over  a  very  long  period  of  years, 
even  if  it  be  not  treated  as  a  debt  in  perpetuity. 

The  argument  as  to  the  large  amount  expended  in  the 
purchase  of  land  is  supported  by  the  substantial  and  permanent 
character  of  the  works  erected  thereon,  and  it  seems  at  first 
sight  sound  policy  to  relieve  the  present  generation  of  rate- 
payers from  what  appears  to  be  an  undue  burden  by  spreading 
the  redemption  of  the  loan  over  a  longer  period  than  is  at 
present  allowed  by  Parliament.  As  against  this  it  is  pointed 
out  that  water  works  have  failed,  water  areas  have  yielded  a 
decreased  and  insufficient  supply,  and  works  which  were  once 
thought  adequate  have,  owing  to  the  large  increase  of  towns, 
become  insufficient  and  have  had  to  be  supplemented  by  further 
outlay.  It  is  also  contended  that  if  the  repayment  of  the  debt 
be  spread  over  a  very  extended  period  the  interest  paid  equals, 
and  soon  exceeds,  the  amount  of  principal.  This  is  not  in 
itself  a  very  good  reason  against  extended  periods  of  repayment 
seeing  that  its  effect  is  to  spread  the  burden  over  a  greater 
number  of  generations  Avho  derive  benefit  from  the  outlay, 
provided  always  that  the  works  continue  to  meet  the  needs  of 
the  community  and  subsequent  generations  do  actually  derive 
a  benefit  therefrom. 

But  it  is  common  knowledge  that  very  few  works  of  public 
utility  last  for  more  than  a  certain  number  of  years.  In  some 
cases  the  rateable  value  of  a  district  falls,  but  in  nearly  all 
cases  the  future  demands  of  the  community  increase  so  rapidly 
that  it  is  imperative  to  put  what  may  by  some  be  termed  an 
undue  burden  upon  the  present  generation  in  order  to  avoid 
placing  an  iutolorablo  l>ur(l('ii   iqioii  the  future.     The  personal 


THE    LIFE    OF    THE    ASSET  383 

element  also  enters  largely  into  the  matter,  and  it  has  been 
found  that  the   surest,   if  not  the  only,  way  to   check   undue 
expenditure,    if    not    extravagance,    upon    the    part    of    local 
authorities  is  to  convert  each  £1,000  of  capital  outlay  into  a 
definite  proportion  of  the  annual  amount  payable  by  the  rate- 
payer by  way  of  rate,  and,  further,  to  educate  the  ratepayer 
to  appreciate  this.     There  is  also  another  interest  to  be  con- 
sidered, namely,  the  loanholder  who  finds  the  money  and  who, 
in    a   great   majority    of    cases,    has    not    any    local    interests. 
He  looks  solely  to  his  security  both  for  the  annual  payment  of 
interest    and    the    ultimate    repayment    of    his    capital.       His 
security  consists  partly  of  the  assets  created  out  of  his  money 
and    partly    of   the    annual    revenues    derived   therefrom,    but 
in     practice     mainly     of     the     future     annual     rates     to     be 
levied    upon   the   community.     Seeing   that   the  value    of    the 
communal  assets  depends  entirely  upon  the  perpetual  prosperous 
existence  of  the  community,  such  assets  have  really  no  value 
unless  the  community  is  able  to  pay  the  future  annual  rates. 
A  bankrupt  or  insolvent  community,   if  not   an   absolute   im- 
possibility,  would  not  be  able  to  pay  any  serious  percentage 
of   its   liabilities;    and   seeing   that   the    security   for  its    loan 
indebtedness  is  a  mixed  fund  of  capital  and  revenue,  of  which 
the  latter  is  the  chief,  it  seems  not  only  reasonable,  but  just, 
that  revenue  or  rate  should  bear  the  greater  proportion  of  the 
burden.     It    follows,    therefore,    that   the    cost    of    the    outlay 
should  be  repaid  within  the  productive  life  of  the  asset  and  be 
charged  against  the  annual  rates  levied  by  the  local  authority. 
In  the  case  of  revenue  earning  undertakings  it  may,  not  very 
unreasonably,   be   contended   that    any   surplus    profits   should 
partially,    if   not   wholly,    be    applied    in   redemption    of   debt 
instead  of  in  aid  of  rate.     If  the  whole  of  such  profits  were 
applied   in   redemption   of   debt,    it   would   avoid    the    present 
anomaly  of  towns  with   equal   annual   rates  but   with  widely 
varying  expenditures,   due  solely  to  the  fact  that  the   excess 
expenditure  in  one  case  is  concealed  by  the  profit  derived  from 
trading  departments.     In  the  case  of  tramways  this  profit  is 
fairly  earned  since  there  is  a  generally  accepted  level  of  fares 
all  over  the  country,  but  in  the  case  of  gas  and  electric  lighting 
undertakings  there  is   such   a  wide   divergence   of   charges   as 
between  different  municipalities,  that  a  very  high  charge,  levied 
at  will  by  the  local  authority,  is  called  a  profit,  and  is  taken 
out  of  the  pockets  of  one  class  of  ratepayers,  namely,  the  gas 
or  electricity  consumers,  and  applied  in  relief  of  the  rates  paid 
bv  the  whole  community. 


384    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

The  foregoing  remarks  deal  very  fully  with  waterworks  as 
representing  a  class  of  outlay  which  lends  itself  most  readily  to 
the  argument  in  favour  of  a  total  abandonment  of  the  annual 
charge  against  revenue  or  rate  in  respect  of  the  redemption  of 
debt,  or  at  least  in  favour  of  a  reduction  in  the  annual  charge 
to   the   present   generation   of   ratepayers   to  the   possible   and 
probable   detriment   of   future   generations.        They   will   have 
their  own  burdens  to  bear  both  as  to  their  then  present,  and 
future  obligations.    Any  relaxation  of  the  present,  as  some  think, 
stringent  regulations  and  practice  will  most  probably  give  them 
in  addition  a  past  burden  to  bear,  which,  owing  to  the  foresight 
of   our   local   authority   forefathers   we   have   escaped.     Conse- 
quently the  modern  Parliamentary  practice  is  right,  namely,  to 
require  the  redemption  of  the  loan  to  be  spread  over  a  period 
well  within  the  life  of  the  asset  created  out  of  the  loan  and  to 
differentiate   between   various   classes   of   outlay   in   fixing   the 
period  to  be  allowed  in  respect  of  each. 

Outlay  on  Manufacturing  Plants.     So  far  the  enquiry 

has  been  confined  to  capital  outlay  upon  public  works  in  which 

the  greater  part  of  the  cost  is  for  land,  which  rarely  depreciates, 

and  very  often  appreciates  in  value,  or  for  buildings  for  which 

a   very    long    life    may    reasonably    be    expected     seeing    that 

judicious  outlay  npon  repairs  will  prolong  the  life  considerably. 

There  is,  however,   a  further  class  of  outlay  of  a  much  more 

complex    nature    where    the    proportion    of    the    original     cost 

attributable  to  land   is   comparatively   small,   and   the   greater 

part  of  the  outlay  is  in  respect  of  buildings,   motive  power, 

plant  and  machinery,  including  in  the  latter  term  everything 

in  the  nature  of  an  engiiie,  gas  making  plant,  tramway  plant, 

electrical   generating   machines   and    all   the   subsidiary   works 

required.     The  necessity  to  exercise  careful  control  over  such 

outlay  arises  from  the  fact  that  as  the  element  of  a  probable 

appreciation    in    value    decreases,    it    is    requisite    to    provide 

for  the  very  opposite   conditions,   namely,   a   probable   fall   in 

value  due  to  two  causes,  first,   a  gradual  wasting  of  the  asset 

due  to  Avear  and  tear  (which  cannot  be  met  by  current  repairs 

and    renewals   charged    to    revenue    account)    and    the    further 

probability  that  future  advances  in  scientific  and  mechanical 

knowledge  may  result  in  the  discovery  of  new  and  improved 

methods  long  before  the  oi'iginal  ])lant,  etc.,   is  worn  out  and 

the    loan    repaid.       Local    authorities    as    well    as    commercial 

concerns  are  here  confronted  with  a  difficult  problem  and  have 

cai(>fullv  to   consider   whether   it    is    advisable   to   discard    the 


THE    LIFE    OF    THE    ASSET  385 

present  obsolete  plant  which  is  costly  in  operation  and  deficient 
in  productive  power,  and  replace  it  with  more  modern  plant, 
relying  npon  the  saving  in  working  charges  and  the  increase  in 
production  to  recoup  the  annual  burden  imposed  by  installing 
the  modern  outfit.  In  such  an  event  there  is  a  wide  difference 
between  the  conditions  existing  in  the  case  of  a  commercial 
undertaking  and  a  local  authority.  A  commercial  undertaking 
may  set  aside  any  part  of  its  profits  and  so  accumulate  a 
reserve  fund  of  unlimited  amount  for  such  a  contingency; 
whereas,  as  a  general  rule,  a  municipality  is  restricted  as  to  the 
amount  which  may  be  so  set  aside  as  a  reserve  fund,  as  dis- 
tinguished from  a  renewals  fund.  If  a  commercial  concern 
requires  to  undertake  outlay  of  this  nature  there  are  not  any 
statutory  or  other  difficulties  in  the  way  provided  the  credit 
of  the  undertaking  is  good ;  and  it  is  not  always  under  any 
obligation  to  set  aside  part  of  the  profits  towards  the  redemption 
of  debt.  On  the  other  hand,  a  local  authority  is  bound  by 
Statute  to  charge  its  annual  revenue  or  rate  account  with  a 
fixed  sum  to  be  applied  in  redeeming  its  loan  indebtedness,  and 
such  obligation  may  not  be  released  without  the  consent  of 
Parliament.  Any  further  borrowing  powers  required  to  replace 
obsolete  assets  or  outlay  before  the  original  loan  is  repaid,  have 
to  be  granted  by  Parliament,  and  very  severe  scrutiny  is  made 
into  all  the  circumstances,  because  the  grant  of  further  powers 
will  lay  a  double  burden  upon  the  community  until  the  original 
loan  is  repaid.  All  this  tends  to  support  the  present  practice 
of  Parliament,  namely,  to  fix  the  period  of  repayment  at  a 
number  of  years  well  within  the  life  of  the  asset;  in  other 
words,  to  make  the  annual  charge  for  the  redemption  of  debt  a 
little  more  than  equivalent  to  the  normal  rate  of  depreciation 
which  would  be  charged  to  revenue,  or  profit  and  loss,  account 
by  a  prudent  trader.  This  practice  supports  the  view  that  in 
the  case  of  a  local  authority  there  is  not  any  necessity,  in 
respect  of  original  outlay,  to  charge  the  revenue  account  with 
depreciation  or  wear  and  tear  in  addition  to  the  sinking  fund 
instalment.  This  question  of  depreciation  (or  wear  and  tear) 
should  be  kept  entirely  distinct  from  the  provision  of  a  general 
reserve  fund  to  make  good  any  capital  losses  due  to  obsolescence, 
or  to  the  provision  of  a  renewals  fund  to  meet  repairs  which 
cannot  be  made  year  by  year,  such  as  the  periodical  relaying 
of  a  tramway  track.  The  statement  that  the  sinking  fund 
instalment  takes  the  place  of  an  annual  charge  for  depreciation 
requires  important  modification  in  one  respect.  It  has  been 
stated  that  an  annual  charge  for  depreciation  may  be  omitted 

A  A 


386    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

ill  the  case  of  original  outlay  only.  The  ease  is  different  when 
once  the  original  loan  has  been  repaid  and  the  asset  becomes 
the  property  of  the  local  authority  free  from  any  debt  and 
without  the  necessity  to  set  aside  any  annual  instalment,  or  to 
pay  any  interest  upon  the  loan.  It  should  here  be  remembered 
that  under  the  sinking  fund  method  of  repayment  both  these 
annual  charges  are  a  burden  upon  the  revenue  or  rate  account 
of  each  year  of  the  repayment  period,  and  that  this  annual 
burden  is  equal  during  the  whole  of  the  period,  seeing  that 
although  part  of  the  loan  may  have  been  repaid  out  of  the 
sinking  fund,  yet  the  interest  upon  the  amount  of  loan  so  repaid 
must  be  charged  to  the  revenue  or  rate  account  and  added  to 
the  fund.  On  the  final  repayment  of  any  loan  the  revenue  or 
rate  account  is  immediately  relieved  of  a  heavy  annual  charge, 
consisting  of  the  instalment  and  interest  upon  the  loan,  and 
the  local  authority  is  in  possession  of  an  undertaking  which  has 
been  provided  out  of  the  revenue  or  rate  of  previous  years,  and 
in  addition  maintained  in  a  state  of  efficiency  by  annual  repairs 
and  renewals,  and  very  possibly  kept  up-to-date  by  improve- 
ments defrayed  by  means  of  additional  charges  to  revenue  or 
rate. 

It  is  therefore  equitable  to  assume  that  it  is  obligatory  upon 
future  generations  of  ratepayers  to  ensure  that  this  asset  shall 
be  maintained  by  them  in  an  efficient  state,  as  far  as  possible, 
but  since  any  expenditure  upon  repairs  and  renewals  cannot 
prevent  a  further  loss  in  value,  such  wastage  should  be  made 
good  bv  charging  future  years  with  an  annual  sum  in  respect 
of  depreciation.  This  is  a  matter  which  is  frequently  over- 
looked, but  it  is  worthy  of  serious  consideration.  When,  in 
spite  of  all  repairs  and  renewals,  the  asset  becomes  valueless, 
or  so  nearly  so  that  it  cannot  be  worked  at  a  profit  or  economi- 
cally, it  must  be  replaced  and  the  depreciation  fund  in  hand 
may  then  be  applied  in  relief  of  the  cost  of  the  new  works, 
leaving  only  the  balance  to  be  raised  by  further  borrowing. 


Section   VII. 

The  Equation  of  the  Period  ol 
Repayment. 


389 


CHAPTEE  XXXII. 

The  equation  or  the  period  of  repayment  of  loans  repayable 

AT    various    dates    WHICH    ARE    REQUIRED    TO    BE    REDEEMED 

on  one  uniform  date  : 

1.  Where  the  loans  are  authorised  in  respect  of 

outlays   of  varying    character,   each   having   a 

different    life    or    PERIOD    OF    CONTINUING    UTILITY 

and  consequent  repayment. 

2.  Where  the  necessity  to  find  the  equated  period 

OF    repayment    arises    on    the    consolidation    of 

EXISTING    loans. 

The  arithmetical  method  of  finding  the  equated  period 

KNOWN  as  the  equation  OF  PAYMENTS,   THE  TRUE  OR  MATHE- 
MATICAL   method;    and    the    error    in    the    generally 

ADOPTED    arithmetical    METHOD. 


The  Necessity  for  the  Equation  of  the  Period  of  Repay- 
ment. In  the  early  days  of  municipal  loans  they  were  relatively 
small  in  amount  as  compared  with  what  they  are  at  the  present 
day,  and  as  a  general  rule  each  loan  was  sanctioned  for  a 
specific  purpose  and  related  to  one  class  of  outlay  only.  When 
a  sanction  or  authorisation  included  several  classes  of  outlay 
a  definite  amount  of  loan  was  authorised,  and  a  definite  period 
was  prescribed,  for  each  class,  carrying  out  the  provision  m 
Section  234(1)  of  the  Public  Health  Act  of  1875,  namely:  — 

"  Money  shall  not  be  borrowed  except  for  permanent  works 
(including  under  this  expression  any  works  of  which  the 
cost  ought,  in  the  opinion  of  the  Local  Government  Board,  to 
be  spread  over  a  term  of  years)." 

Under  this  Act  (Sec.  234  [4])  the  period  of  repayment  may 
be  fixed  by  the  local  authority  with  the  sanction  of  the  Local 
Government  Board. 

Section  243  of  the  same  Act  dealing  with  loans  to  local 
authorities  by  the  Public  Works  Loan  Commissioners,  provides  : 


390    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

"  That  in  determiuing  the  time  when  a  loan  under  this 
section  shall  be  repa^-able  the  Local  Government  Board  shall 
have  regard  to  the  probable  duration  and  continuing  utility 
of  the  works  in  respect  of  which  the  same  is  required." 

With  the  widening  of  the  sphere  of  municipal  activity  to 
include  gas  works,  tramways,  electric  supply,  hydraulic  power 
supply  and  other  manufacturing  (and  in  many  cases  profit 
earning)  utilities,  the  problem  became  more  complicated,  seeing 
that  the  total  loan  authorised  for  any  one  undertaking  neces- 
sarily included  outlays  of  very  diverse  character  having  widely 
varying  periods  of  utility  and  consequently  varying  periods  of 
repayment.  Further  difficulties  were  introduced,  when,  under 
the  Public  Health  Acts  Amendment  Act,  1890,  local  authorities 
generally  were  empowered,  subject  to  certain  conditions  laid 
down  in  the  Stock  Hegulations  of  1891,  etc.,  to  raise  money 
by  the  issue  of  stock  redeemable  on  a  specified  date  or  at  the 
end  of  a  given  number  of  years. 

During  the  earlier  years,  when  each  loan  was  authorised 
for  one  class  of  outlay  only  with  a  definite  repayment  period, 
all  that  was  necessary  was  to  keep  a  separate  sinking  fund  for 
each  loan,  when  the  whole  amount  was  borrowed  in  one  year, 
and  to  keep  a  separate  fund  for  each  year's  borrowings,  when 
the  loan  was  borrowed  over  a  series  of  years.  The  same  applied 
to  loans  authorised  for  one  undertaking  including  various 
classes  of  outlay,  each  having  a  difl:erent  period  of  repayment, 
so  long  as  the  sinking  funds  could  be  kept  distinct  for  each  class 
of  outlay  or  each  year's  borrowings,  and  the  funds  could  mature 
at  the  end  of  the  respective  periods  and  the  loans  then  be 
redeemed.  But  when  it  became  possible  to  raise  loans  by  the 
issue  of  stock  redeemable  on  a  fixed  date  it  at  the  same  time 
became  necessary  to  so  arrange  the  sinking  fund  instalments 
that  the  total  loan  should  be  redeemed  on  the  prescribed  date 
irrespective  of  tlie  repayment  periods  imposed  for  the  several 
component  parts  of  the  outlay. 

The  difficulty  is  overcome  by  ascertaining  the  equivalent 
average  date  of  repayment  of  the  Avhole  of  the  loan,  and 
calculating  the  annual  instalment  required  to  be  set  aside  and 
accumulated  in  one  instead  of  in  several  sinking  funds.  The 
actual  practice  varies.  In  some  cases  the  sanction  states  the 
specific  amounts  to  be  borrowed  for  each  class  of  outlay  with 
the  period  of  repayment  allowed  for  each  class,  and  the  duty 
of  fixing  the  average  date  falls  upon  the  local  authority.     In 


THE  EQUATION  OF  THE  PERIOD  OF  REPAYMENT       391 

other  cases  the  local  authority  submits  a  scheme  to  the  Local 
Government  Board  showing  the  various  sums  proposed  to  be 
borrowed  for  each  class  of  outlay,  the  respective  periods  of 
repayment  suggested  and  the  proposed  average  date  of  repay- 
ment of  the  whole  loan.  This  is  subject  to  revision  by  the 
Government  department  concerned,  especially  as  to  the  period 
desired  by  the  local  authority,  and  this  being  fixed  the  average 
or  equated  period  is  found  by  calculation  in  a  manner  which 
will  be  discussed  in  detail. 

These  are  the  general  considerations  which  are  involved  in 
the  equation  of  the  period  of  repayment,  but  they  may  be 
further  complicated  by  reason  that  the  amounts  are  borrowed 
over  a  series  of  years,  or  that  the  loans  in  respect  of  the 
component  parts  of  the  outlay  are  borrowed  together  at 
irregular  times,  and  without  any  definite  allocation  as  between 
the  various  classes  of  outlay.  In  many  cases  it  is  necessary 
to  set  aside  temporary  instalments  during  construction,  leaving 
the  final  instalment  to  be  ascertained  by  adjustment  when  the 
total  loan  has  been  borrowed  and  the  whole  of  the  works  carried 
out  and  an  apportionment  made  of  the  outlay.  In  the  case  of 
very  large  undertakings  this  cannot  be  done  until  the  engineer 
has  given  his  final  certificate. 

Another  ditficulty  arises  in  cases  where  the  operation  of  the 
sinking  fund  is  suspended  for  a  number  of  years,  and  it  is  often 
almost  impossible,  owing  to  a  combination  of  the  above  factors, 
to  decide  upon  the  amount  of  the  first  instalment  to  be  set  aside. 
The  only  permanent  factor  is  the  repayment  of  the  whole  of  the 
loan  on  a  fixed  date.  The  same  considerations  apply  on  the 
consolidation  of  several  existing  loans  repayable  at  various 
dates,  when  it  becomes  necessary  to  fix  a  uniform  date  of 
repayment  and  adjust  the  instalment,  having  regard  to  the 
amounts  now  in  the  several  sinking  funds. 

Space  will  not  permit  of  the  detailed  treatment  of  any  such 
examples  owing  to  the  difficulty  of  stating  a  set  of  conditions 
which  would  be  generally  applicable.  Each  case  must  be 
dealt  with  on  the  individual  facts,  but  any  question  likely  to 
arise  may  be  treated  by  one  or  more  of  the  methods  described 
in  this  book.  As  a  general  rule,  where  the  conditions  are  at  all 
complicated,  it  is  better  to  set  aside,  during  the  construction 
period,  temporary  instalments  of  a  general  natiire  and  defer 
any  final  adjustment  until  the  whole  of  the  loan  has  been 
borrowed  and  the  actual  outlay  under  each  head  has  been 
certified  by  the  engineer. 


392         REPAYMENT   OF   LOCAL   AND   OTHER   LOANvS 

The  Methods  of  Finding  the  Equated  Peeiod  of  Repay- 
ment. Tlie  foregoing  remarks  will  uow  be  illustrated  by  tbe 
following  example  whicli  is  of  a  simple  character  witliout  any 
of  the  complications  previously  referred  to,  and  relates  to 
a  loan  of  £56,000,  raised  by  the  issue  of  stock  redeemable  at 
par  in  one  sum  on  a  date  to  be  ascertained.  The  loan  is 
required  for  an  undertaking  comprising  outlay  of  a  variable 
nature,  each  class  of  Avliich  has  a  different  life  or  period  of 
utility,  and  separate  periods  are  prescribed  for  each.  In  order 
to  ascertain  the  date  of  redemption  of  the  stock  it  is  required 
to  find  the  equated  period  corresponding  to  the  several  prescribed 
periods  and  amounts.  This  method  will  apply  to  ordinary 
loans  if  it  be  required  to  repay  the  total  debt  on  one  date. 

The  classes  of  outlay,  the  amounts  of  loan  authorised  in 
respect  of  each  class  and  the  prescribed  periods  of  repayment 
are  as  follows :  the  rate  of  accumulation  is  8  per  cent,  per 
annum.  The  rate  of  interest  payable  upon  the  stock  does  not 
enter   into   the    calculation. 

TABLE  XXXII.  A. 

Particulars  of  the  Loan  of  £56,000. 


Nature  of  outlay. 

Amount  of  loan 
authorised. 

Prescribed  period. 

Class  A 

£10,000 

repayable  in  45  years 

55         13 

20,000 

on 

)5                             55      ~''^               5  5 

,5         C 

24,000 

55                             55       1^              55 

55        ^ 

2,000 

55                             55           '^              55 

£56,000 


The  Arithmetical  Method.  Under  ordinary  circum- 
stances the  above  amounts  of  loan  woiild  be  repayable  at  the 
end  of  the  respective  periods  by  means  of  the  usual  sinking 
fund  instalments,  as  described  in  previous  chapters,  but  under 
the  present  conditions  the  whole  of  the  loan  is  repayable  on 
one  date,  which  has  to  be  so  fixed  that  the  lender  will  receive 
his  money  at  a  time  equivalent  to  that  at  which  he  would  have 
received  it  if  the  original  varying  periods  and  amounts  had 
been  adhered  to.  In  arithmetic  this  is  known  as  "  the  equation 
of  payments,"  and  the  rule  is  stated  as  follows  :  — 

A/ulfiply   each  debt  by   tJie   number   of  years   which    will 

elapse  before  it  becomes  payable;  add  the  results  together; 

divide  this  sum.  by  the  sum  of  the  debts;  the  quotient  will 

be  the  number  of  years  in  the  equated  time. 


THE  EQUATION  OF  THE  PERIOD  OF  REPAYMENT      393 

But  it  is  stated  in  tlie  books  on  arithmetic  that  this  is  only 
approximately  correct,  and  can  only  be  taken  as  equitable  when 
the  various  times  of  repayment  are  not  widely  apart.  The 
error,  it  is  pointed  out,  is  in  favour  of  the  payer  as  it  extends 
the  period  of  repayment.  This  arithmetical  method  will  first 
be  applied  to  ascertain  the  equated  period  of  repayment  of  the 
above  loan  of  £56,000,  after  which  an  investigation  will  be 
made  in  order  to  ascertain  the  true  equated  period  suggested  in 
the  arithmetic  book.  This  is  the  more  necessary  because  in 
the  case  of  the  loans  of  local  authorities  the  various  times  of 
repayment  are  very  widely  apart.  The  result  of  the  investiga- 
tion into  the  true  equated  period  will  show  that  it  is  shorter 
than  under  the  arithmetical  method.  In  the  case  of  a  local 
authority,  however,  the  arithmetically  equated  period  may  be 
preferred  because  it  is  slightly  in  favour  of  the  payer  (in  this 
case  the  revenue  or  rate  account  of  the  equated  period)  as  it 
extends  the  repayment  beyond  the  time  required  by  the  true 
equated  method.  The  effect  of  equating  several  sinking  fund 
periods  is  to  reduce  the  total  period  over  which  the  repayment 
is  spread  and  thereby  relieve  part  of  the  original  period  of  any 
charge  whatever.  The  burden  of  this  relief  is  thrown  upon  the 
equated  period  taken  as  a  whole,  and  any  extension  of  this 
period  tends  to  redress  the  inequality  caused  by  the  equation. 
With  regard  to  the  interest  upon  the  loan,  which  will  be 
considered  fully  in  Chapter  XX  XY,  it  should  be  remembered 
that  under  the  original  conditions  the  annual  interest  charge 
to  revenue  or  rate  will  gradually  be  reduced  as  the  loans  with 
shorter  prescribed  periods  are  repaid;  whereas  under  the 
generally  adopted  method  of  distributing  the  annual  burden 
after  equation,  interest  upon  the  full  amount  of  the  loan  is 
payable  equally  during  and  charged  equally  against  the  revenue 
or  rate  account  of  each  year  of  the  equated  period.  This  will 
be  discussed  in  a  later  chapter,  but  the  present  subject  of 
enquiry  relates  solely  to  the  method  of  finding  the  true  e([uated 
period. 

The  calculation  of  the  equated  period  relating  to  the  above 
loan  of  £56,000  will  now  be  made,  by  the  arithmetical  rule, 
and,  although  the  figures  adopted  give  a  period  of  an  exact 
number  of  years,  yet  in  practice  this  will  rarely  be  obtained. 
It  is  in  fact  somewhat  difficult  to  state  original  conditions 
which  will  work  out  to  an  even  number  of  years  in  the 
equated   period. 


394         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 

TABLE  XXXII.  B. 

The  Arithmetical  Method  of  Finding  the  Equated  Period 
or   Repayment. 

Required  the  equated  period,  at  the  end  of  wliicb  the  total 
loan  should  be  repayable,  corresponding  to  the  repayment 
of  the  component  parts  of  the  loan  at  the  end  of  the 
respective  periods  prescribed  for  each. 


iture  of  outlay. 

Amount  of 
loan  authorised. 

Prescribed 
periods. 

Product  of  amount 

of  loan  multiplied  by 

number  of  years. 

Class    A 

£10,000 

45  years 

£450,000 

„      B 

20,000 

^9      ,, 

580,000 

„      c 

24,000 

15     „ 

360,000 

„       B 

2,000 

5     „ 

10,000 

£56,000 

£1,400,000 

Equated  period  :  — 
1,400,000  _ 


25  years. 


56,000 

The  Trie  or  Mathematical  Method.  The  correctness  of 
the  above  arithmetically  equated  period  will  now  be  investi- 
gated, as  well  as  the  eifect  of  the  alteration  upon  the  repayment 
of  the  loans. 

To  do  this  it  is  first  necessary  to  ascertain  the  exact 
equivalent  of  the  original  conditions.  This  will  be  stated  as 
if  those  conditions  had  been  carried  out  hj  setting  aside  an 
equal  annual  instalment  in  respect  of  each  of  the  amounts  of 
loan  and  accumulating  them  in  four  separate  sinking  funds  to 
repay  the  several  portions  of  the  loan  at  the  end  of  5,  15,  29, 
and  45  years  respectively.  But  as  the  individual  sinking  funds 
mature  at  different  dates  each  annual  instalment  must  be 
reduced  to  its  present  value.  The  sum  of  such  present  values 
represents  the  amount  of  money  noAv  required  to  purchase  an 
equivalent  annuity  or  annual  instalment  for  the  equated  period 
of  25  years.  For  purposes  of  the  comparison  to  be  made  in 
Chapters  XXXIV  and  XXXV  it  is  necessary  to  know  the 
individual  instalments  to  be  set  aside  during  the  whole  of  the 
above  periods,  and  tlie  annual  instalment  as  well  as  its  present 
value  will  therefore  be  shown  in  each  case.  The  calculations 
are  all  similar  to  others  which  have  been  previously  worked  out 
so  that  it  is  not  necessary  to  show  the  actual  working  as  in 
earlier  chapters. 


THE  EQUATION  OF  THE  PERIOD  OF  REPAYMENT       395 

TABLE  XXXII.  C. 

The       equation      of      the      I'EKIOD     of      RErAYMENT      OF       LOANS, 

repayable  at  various  dates,  wliicli  are  required  to  be  re- 
deemed on  one  uniform  date. 

Loan  of  £56,000,  authorised  for  outlays  of  varying  cliaracter, 
each  having  a  different  life  or  period  of  continuing  utility, 
and  a  consequent  period  of  repayment. 

Eate  of  accumulation,  3  per  cent. 

Annual  instalments  required  under  the  original  conditions. 

Equated  period  for  a  loan  for  public  works  consisting  of  outlay 
having  varying  lives  or  periods  of  continuing  utility. 


Period 

Nature  of  allowed  for 

outlay.  repayment.  Details 

Class  A  45  years  — 

„      B  29     „  - 

,,      C  15     „  — 

..      D       5     ,.  — 


Amount  of  loan  Present  value  of 

authorised.                  Annual  Loan                  Future 

instalment  repayable  at            annual 

Total.          to  repay  loan,  end  of  period.      instalments. 


10,000  107-85  2644-39  264439 

20,000  442-29  8486-93  8486-93 

24,000  1290-40  1540469  15404-69 

2,000  376-71  1725-22  1725-22 


Total      45  years     —     56,000     2217-25     28261-23     28261-23 


The  present  values  in  the  above  table  are  the  present  values 
both  of  the  amounts  of  loan  repayable  at  the  end  of  the 
respective  periods  and  also  of  the  corresponding  sinking  fund 
instalments,  since  the  instalments,  if  accumulated,  will,  at  the 
end  of  the  respective  periods,  amount  to  the  respective  loans. 

The  whole  of  the  loans,  although  repayable  at  the  end  of 
successive  periods,  have  now  been  reduced  to  a  common 
measure,  namely,  a  "  present  value "  of  £28261-23,  which 
represents  the  amount  for  which  the  various  sinking  fund 
obligations  might  be  redeemed  at  the  present  time,  and  upon 
which  the  calculation  of  the  true  equated  period  will  be  based. 

The  folloAving  argument  is  summarised  in  Table  XXXII.  D., 
which  may  be  referred  to  with  advantage  :  — 

If  the  arithmetical  calculation  of  the  equated  period  of  25 
years  be  correct  this  sum  of  £28261-23  should  in  25  years,  at 
3  per  cent.,  amount  to  £56,000,  and  the  annuity  which  it  will 
purchase  (or  the  sinking  fund  instalment)  should  also  amount 
to   £56,000   in   that   period.     This,    however,    is   not   the   case. 


396         REPAYMENT    OF    LOCAL   AND    OTHER   LOANS 

It  may  be  found  by  calculation  that  i>28261"23  will  in  25  years, 
at  3  per  cent.,  amount  to  £591T2"75,  or  an  excess  of  £3172'75. 
And  it  may  also  be  ascertained  that  the  annuity  or  annual 
instalment,  which  will  amount  to  £591 72' 75  in  25  years,  at 
3  per  cent.,  and  of  which  £28261'23  is  the  present  value,  is 
£1622"98  per  annum.  Since  £28261'23  is  also  the  present  value 
of  the  four  annual  instalments  required  to  provide  the  com- 
ponent parts  of  the  loan  of  £56,000  at  the  end  of  the  respective 
periods  of  5,  15,  29  and  45  years,  it  is  obvious  that  the  error 
lies  in  the  number  of  years  in  the  equated  period,  as  found  by 
the  arithmetical  method. 

The  next  step  is  to  calculate  the  actual  annual  instalment 
(and  also  the  present  value  of  the  instalment)  required  to  repay 
£56,000  in  25  years,  the  equated  period  as  found  by  the 
arithmetical  method,  but  which  there  is  reason  to  suspect  is  in 
excess  of  the  true  period.  It  will  be  found,  on  making  the 
calculation,  that  the  annual  instalment  to  repay  £56,000  in 
25  years  is  £153595,  and  its  present  value  £2674590  (which  is 
also  the  present  value  of  £56,000  due  at  the  end  of  25  years). 
This  annual  instalment  of  £1535'95  cannot,  of  course,  be 
compared  with  the  four  sinking  fund  instalments,  amounting 
together  to  £221725,  to  be  accumulated  for  the  original  periods 
because  they  are  all  for  different  numbers  of  years,  but  it  has 
been  ascertained  that  they  are  equivalent  to  an  annual  instal- 
ment of  £162298  to  be  set  aside  for  25  years,  and  accumulated 
at  3  per  cent.  It  is  therefore  possible  to  compare  the  two 
annual  instalments  of  £153595  and  £1622"98,  and  the  result  is 
to  prove  that  the  arithmetically  equated  period  gives  an  annual 
instalment  which  is  less  by  £87  03  than  the  exact  equivalent  of 
the  original  instalments.  In  other  words,  the  arithmetically 
equated  period  is  too  long.  The  following  table  (XXXII.  D.) 
shows  the  above  conclusions  :  — 


TABLE  XXXII.  D. 

The  True  or  Mathematical  Method  of  Finding  the  Equated 
Period  of  Repayment, 

Showing  the  annual  instalments  and  their  present  values  under 

(1)  the  original  conditions; 

(2)  the  arithmetically  equated  period ; 

(3)  the  true  or  mathematically  equated  period. 


THE  EQUATION  OF  THE  PERIOD  OF  REPAYMENT      397 


Sinking  fund  instalment  ami 
present  value  thereof  for 


Original      periods      of 
repaymejit    


Amount 
of  loan. 

10,000 

20,000 

24,000 

2,000 


Sinking  fund 
instalment 
Number  per 

ofj'ears.  annum. 


Present 

value  of 

loan  or 

sinking  fund 

instalment. 


45  107-85  2644-39 

29  442-29  8486-93 

15  1290-40  15404-69 

5  376-71  1725-22 


56,000 

2217-25 

28261-23 

25   years,    the    equated 
period  as  found  by 
the        arithmetical 

method  :  — 
Amount  of  loan 

5600000 

25 

1535-95 

26745-90 

Amovmt  which  will  be 

provided  at  end  of 
25th  year     

59172-75 

25 

1622-98 

28261-23 

Surplus     

3172-75 

25 

87-03 

1515-33 

24  years : — 

Amount  of  loan      ... 

56,000 

24 

1626-65 

27548-28 

23  years : — 

Amount  of  loan 

56,000 

23 

1725-58 

28374-73 

Amount  which  will  be 

provided  at  end  of 
23rd  year     

55,776 

23 

1718-68 

28261-23 

Deficiency 

224 

23 

6-90 

113-50 

Having  ascertained  the  exact  error  in  the  annual  instalment 
under  the  arithmetical  method  of  equation  the  error  in  the 
equated  period  itself  may  now  be  found.  It  has  already  been 
ascertained  that  at  3  per  cent.  £26745-90  will  amount  to  £56,000 
in  25  years,  and  the  problem  is  to  ascertain  in  how  many  years 
£28261-23  will  amount  to  £56,000.  As  the  present  value  of 
the  four  original  instalments,  namely,  £28261*23,  is  greater 
than  £26745-90,  which  is  the  present  value  of  £56,000,  it  will 
amount  to  £56,000  in  a  smaller  number  of  years  than  25.  The 
exact  number  of  years  may  be  ascertained  by  using  the  formula 
relating  to  Table  I,  in  standard  calculation  form,  No.  1,  to  find 
the  amount  of  £1  in  any  number  of  years,  but  this  will  give 
a  result  consisting  of  a  number  and  a  fractiou.     In  cases  such 


398    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

as  tlie  present  the  exact  fractiou  of  the  year  is  required  only 
for  the  purpose  of  fixing-  the  nearest  number  of  whole  years, 
so  that  the  problem  will  work  out  in  practice.  The  method  of 
finding  the  number  of  years  by  using  the  formula  relating  to 
Table  I  is  as  follows,  and  may  be  compared  with  the  standard 
calculation  form  for  the  purpose  given  in  Chapter  X. 

STATEMENT  XXXII.  E. 

Required  the  number  of  years  in  which  £28261-23  will  amount 
to  £56,000  at  3  per  cent. 

By  formula  and  logs. 

A  =  PEN  56,000  =  28261-23  x  103N. 

Log.  amount  at  end  of  period       ...         56,000  4-7481880 

deduct  Log.  present  value       ...         2826r23         4-4511911 

=  Loo-   RN      ...  0-2969969 


divide  bv  Log.   E=  1-03       ...  00128372 


To  divide  one  Log.  by  another  find 

the(Logs.  of  the  above  Logs,  as  if 

they  were  actual  numbers,  viz.  : 

Log.  2969969  =  6-4727516 

Los-.    128372  =  5-1083703 


1-3642813 


which  is  the  Log.  of  the  number  of  years,  viz.  ...  23-136 

In  order  to  avoid  the  necessity  of  dividing  one  log  by 
another,  the  exact  number  of  years  may  be  ascertained  by 
means  of  Thoman's  Table  giving  the  logs  of  R^,  at  3  per  cent., 
for  various  years,  as  follows:  — 

Proceed  as  in  the  above  Statement  by  deducting 
the  log  of  the  present  value  of  the  annual 
instalments  under  the  original  conditions, 
from  the  log  of  the  amount  of  loan  repay- 
able at  the  end  of  the  period.  The  remain- 
der is  the  log  of  RN,  from  wliich  the  value 
of  N,  mav  be  obtained.  In  the  above  case, 
the  loc.  of  RN  is     0-2969969 


THE  EQUATION  OF  THE  PERIOD  OF  REPAYMENT      399 

On  referring  to  Thoman's  Table,  tlie  nearest  logs  of  E^,  above 
and  below  this,  are  found  to  be  as  follows  :  — 

at  ;}  per  cent.,  log  RN     24  years    0-30809;J4 

23  years    0-2952562 


a  difference  of    0-0128372 


The  next  step  is  to  find  the  difference  between  the  log.  of  H^, 
as  found  in  the  above  calculation  in  Statement  XXXII.  E.,  and 
from  which  it  is  required  to  find  the  value  of  N ;  and  the  lower 
of  the  above  logs  in  Thoman's  Table  as  follows:  — 

log  of  EN^  ill  calculation,  as  above     0-2969969 

log  of  EN    23  years,  by  Thonian     02952562 

a  difference  of    0-0017407 


and  the  fraction  of  a  year  above  23  years  is  :  — 

—  or  0- 135598  as  may  be  found  by  logs. 


128372 


The  number  of  years  therefore  is  23-136,  and  agrees  with  the 
calculation  in  Table  XXXII.  E.,  made  by  means  of  the  formula. 
Another  method  of  making  the  calculation,  after  having 
found  the  above  difference  of  0-0128372  in  the  logs,  is  to  refer 
to  the  tables  of  differences  given  in  the  margin  of  the  ordinary 
log  tables,  and  under  128  the  following  amounts  will  be  found  : 

•10  =0-12800 
•07  =0-00900 
•004  =  000051 


174  =  013751  of  1  year, 


which  differs  from  the  previous  result  by  less  than  1  day. 

Summary  of  the  True  or  Mathematical  Method.  In 
order  to  ascertain  the  number  of  years  in  the  true  equated 
period  it  is  advisable  to  find,  first,  the  approximate  number 
of  years  by  the  arithmetical  method,  in  this  case  25  years,  and 
then  to  find  by  calculation  in  the  manner  already  described, 
and  shown  in  Table  XXXII.  C,  the  present  value  of  the  several 
annual  instalments  under  the  original  conditions  before 
equation ;  in  the  present  instance,  £28261-23.     The  next  step  is 


400         REPAYMENT   OF   LOCAL   AND    OTHER   LOANvS 

to  find  the  amount  of  loan,  £59172"T5,  wliicli  will  be  provided 
by  the  accumulation,  at  Ibe  estimated  rate,  of  the  above  present 
value,  for  a  number  of  years  (25)  equal  to  the  equated  period, 
as  ascertained  by  the  arithmetical  method.  The  amount  of 
loan  which  will  be  thereby  provided  should  then  be  compared 
with  the  actual  amount  of  the  loan.  As  a  general  rule  the 
amount  of  loan  Avhich  will  be  provided  by  the  accumulation  of 
the  present  value  of  the  annual  instalments  under  tlie  original 
conditions  before  equation  at  the  end  of  the  equated  period, 
will  be  greater  than  the  amount  of  the  loan,  and  will  denote 
that  the  equated  period  as  found  by  the  arithmetical  method 
is  in  excess  of  the  true  equated  period. 

The  enquiry  is  therefore  confined  to  the  present  value  of  the 
actual  loan,  at  the  estimated  rate  of  accumulation,  for  periods 
of  years  less  than  the  number  of  years  (25)  as  found  by  the 
arithmetical  method. 

Eeference  is  nest  made  to  the  tables  of  compound  interest  in 
order  to  ascertain  the  present  value  of  the  loan  at  the  estimated 
rate  of  accumulation  for  periods  less  than  the  arithmetically 
equated  period. 

Reverting  to  the  present  example,  a  period  of  24  years  will 
first  be  taken,  and  it  will  be  found  by  calculation  on  standard 
calculation  form,  No.  2,  that  the  present  value  of  £56,000  due 
at  the  end  of  24  years  is  £27548'28,  requiring  an  aunual  instal- 
ment (as  may  be  found  by  standard  calculation  form,  No.  8x)  of 
£1626-65. 

The  above  present  value,  £27548"28,  as  compared  with 
£2826r23,  the  actual  present  value  of  the  original  annual 
instalments,  before  equation,  is  still  insufficient,  and  a  period 
of  23  years  is  adopted.  Similar  calculations  will  show  that 
the  present  value  of  £56,000  due  at  the  end  of  23  years,  is 
£28374*73,  which  is  very  nearly  correct.  And  therefore  23  years 
is  adopted  as  the  nearest  to  the  true  equated  period.  The 
future  annual  instalment  to  be  spread  equally  over  the  equated 
period  may  now  be  ascertained  by  calculation  on  standard  form 
3x,  and  will  be  found  to  be  £1725-58. 

The  only  conchislon  which  may  properly  be  drawn  from 
the  above  facts,  is,  that  an  annual  instalment  of  £1725"58  to  be 
accumulated  for  23  years  at  3  per  cent.,  is,  within  a  small  limit 
of  error,  the  true  mathematical  equivalent  of  the  four  annual 
instalments  under  the  original  conditions,  amouutiug  together 
to  £2217-25,  as  shoAvn  in  Table  XXXII.  C,  to  be  accumulated 
for  the  respective  periods  shown  in  that  table.  It  has  nothing 
whatever   to   do   with    tlie    incidence   of   Ibe   bnrden    upon    the 


THE  EQUATION  OF  THE  PERIOD  OF  REPAYMENT      401 

revenue  or  rate  accounts  of  the  equated  period,  wliicli  will  be 
fully  considered  in  a  later  chapter. 

The  correct  figures  as  to  the  equated  period  of  repayment 
of  the  above  loan  are  therefore  as  follows  :  — 

Amount  of  loan  repayable  at  the  end  of  23  years       £56,000 
Present  value  thereof £283T4"73 


Annual  instalment  of  which  £28374'73  is  the 
present  value,  and  which  will  amount  to 
£56,000  in  23  years  at  3  per  cent 


£1725-58 


Owing  to  the  fact  that  the  equated  period  is  fixed  at  the  nearest 
whole  number  of  23  years,  instead  of  23"  136  years,  as  shown  in 
Statement  XXXII.  E.,  the  annual  instalment  of  £1725-58  is 
larger  than  the  instalment  £1718-68,  which  is  the  equivalent 
of  the  present  value  £28261-23,  of  the  original  instalments 
shown  in  Table  XXXII.  C.  The  following  table  shows  the 
error  involved  bv  taking  the  nearest  whole  number  of  years  :  — 


23  years  3  per  cent, 
based  upon  : — 

Actual  amount  of  loan 

Actual   j^resent  value 

of  the  original  in- 


Capital 


Present 
value. 


Annual 
instalment. 


stalments 


£56,000         £28374-73         £1725-58 


£55,776        £28261-23        £1718-68 


Difference 


£224 


£113-50 


£6-90 


The  above  table  shows  that  £28261-23  will  not  amount  to 
£56,000  in  23  years  but  only  to  £55,776,  requiring  an  annual 
instalment  of  £1718-68,  consequently  it  is  not  possible  to  arrive 
at  anything  nearer  than  an  approximation  of  the  period.  The 
annual  instalment  to  be  set  aside  for  23  years  corresponding  to 
the  present  value  of  £28261*23,  namely  £1718*68,  is  less  by 
£6-90  than  the  instalment  required  to  repay  £56,000,  and  would 
fall  short  of  repaying  the  loan  by  £224  at  the  end  of  23  years. 
The  method  is  an  approximation  only  and  in  actual  practice 
the  arithmetical  method  would  give  a  number  of  years  contain- 
ing a  fraction,  but  the  result  is  sufficientlv  correct  if  the  nearest 
number  of  even  years  be  taken. 

The  effect  of  adopting  an  equated  period  of  25  years,  as 
shown  by  the  arithmetical  method,  instead  of  23  years  as  shown 


AB 


402  REPAYMICNT    OF    LOCAL   AND    OTHER   LOANS 

by  the  true  equated  method,  may  be  seen  from  an  inspection 
of  Table  XXXII.  D.  It  may  be  taken  as  a  general  rule  that 
the  arithmetical  method  gives  the  longer  repayment  period, 
and  relieves  the  revenue  or  rate  accounts  of  the  equated  period 
as  compared  with  the  true  method  of  equation  which  should 
always  be  used  when  it  is  desired  to  accelerate  the  repayment 
of  the  loan,  or  when  extreme  accuracy  is  required. 

Further  Proof.  The  previous  example  of  the  equation  of 
the  period  of  repayment  of  a  loan  of  £56,000  is  not  a  case 
occurring  in  actual  practice.  The  amounts  of  outlay  composing 
the  loan  as  well  as  the  periods  of  repayment  are  all  assumed, 
and  the  problem  has  been  treated  purely  from  the  theoretical 
standpoint  in  order  to  show  the  difference  between  the 
arithmetical  and  true  methods  of  finding  the  e(|uated  period. 
The  basis  of  the  method  there  adopted  is  to  ascertain,  first, 
the  annual  instalments  required  in  respect  of  each  part 
of  the  loan  and  then  to  find  the  present  value  of  such  instal- 
ments. The  same  present  values  may  be  obtained  in  one 
operation  by  finding  the  present  values  of  the  several  amounts 
of  loan  repayable  at  the  ends  of  the  respective  periods,  where  it 
is  not  necessary  to  know  the  actual  instalments,  as  was  the  case 
in  the  foregoing  example. 

A  further  example  will  now  be  given,  using  figures  occurring 
in  actual  practice,  but  with  a  shorter  period  of  ultimate  repay- 
ment than  45  years,  and  the  results  will  be  compared  with  the 
previous  example.  The  calculation  will  be  made  by  the  shortest 
possible  method.  The  actual  example  may  be  found  in  the 
report  by  a  Select  Committee  of  the  House  of  Commons  upon 
the  Repayment  of  Loans  by  Local  Authorities  (1902),  page  261. 
This  example  of  an  equated  period  Avas  put  in  evidence  by  the 
Assistant  Secretary  of  the  Local  Government  Board  to  illustrate 
the  method  adopted  by  the  lioard  in  order  to  arrive  at  the 
average  period  to  be  granted  for  the  repayment  of  a  loan  to  be 
expended  upon  a  gas  undertaking  where  the  component  parts  of 
the  outlay  have  a  variable  probable  duration  and  continuing 
utility. 

The  following  table  shows,  in  the  first  four  columns,  the 
nature  of  the  outlay,  the  period  allowed  for  repayment  in 
respect  of  each,  and  the  amount  of  loan  to  be  expended  in  each 
case.  The  fourth  column  shows  the  component  parts  of  the 
loan  in  respect  of  which  the  same  period  is  allowed.  The 
details  arc  taken  from  the  appendix  to  the  above  report.  In 
ordcT-  to   avoid    repetition    tlie  table   also   contains  the   present 


THE  EQUATION  OF  THE  PERIOD  OF  REPAYMENT      403 

values  of  the  component  parts  of  tlie  loan  as  found  by 
calculation.  The  annual  instalments  are  not  shown,  because  in 
this  case  they  do  not  enter  into  the  calculation.  This  table  may 
be  compared  with  Table  XXXII.  C. 


TABLE  XXXII.  F. 

The  Equation  of  the  Period  of  Repayment  of  Loans 
repayable  at  various  dates,  which  are  required  to  be 
redeemed  on  one  uniform  date. 

Loan  of  £9105,  authorised  for  outlays  of  varying  character, 
each  having  a  different  life,  or  period  of  continuing  utility, 
and  a  consequent  period  of  repayment. 

Rate  of  accumulation,  3  per  cent. 

Equated  period  for  a  loan  for  Gas-works  purposes. 


Present  value  of 

Nature  of  outlay. 

Period 
allowed 

for 
repayment. 

Amount  of  loan 
authorised. 

Details.              Total. 

Annual 

instalment 

to  repay 

loan. 

Loan 
repayable  at 
end  of  period. 

Future 
annual 

instalments. 

Buildings 

30 

years 

250000 

.siii 

Mains 

jj 

124500 

Gasometer 

35 

150000 

lof  1 
1   is 
)f  th 
nt  V 

Condensers 

jj 

530-00 

2.2^1   . 
^  ^  3  *-<  -*^ 

30 

years 

57T500 

2379-22 

2379-22 

Purifiers 

20 

years 

100000 

^   3    r/;   0   ^ 

553-67 

553-67 

Benches 

15 

5J 

120000 

°  "  -^-3 

770-23 

770-23 

Meters 

10 

5> 

530-00 

this 
true 
1  the 
ad  c 
,nnu 

394-37 

394-37 

Retorts 

2 

)J 

600-00 

In 

the 
upon 
inste 
the  a 

565-55 

565-55 

30 

years 

9105-00 

4663-04 

4663-04 

The  report  shows  also  the  arithmetical  method  adopted  to 
arrive  at  the  equated  period  of  repayment  of  the  total  loan 
authorised,  and  this  method  corresponds  exactly  with  the 
method  laid  down  in  the  books  on  arithmetic  and  illustrated 
by  Table  XXXII.  B.  The  actual  working  is  given  in  the 
report  and  may  be  summarised  as  folloAvs  :  — 


404         REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 


TABLE  XXXII.  G. 


The  Arithmetical  Method  of  Finding  the  Eqlated  Peeiod 
OF  Eepayment  of  the  Lo.\n. 


Nature  of  outlay. 

Amount 

of  loan 

authorised. 

Prescribed 
periods. 

Product  of  amount 
of  loan  multiijlied 

by 
number  of  years. 

Buildings,  etc. 

5,775 

30  years 

173,250 

Purifiers 

1,000 

20     „ 

20,000 

Benches 

1,200 

15     ,, 

18,000 

Meters      

530 

10     „ 

5,300 

Retorts     

600 

1,200 

Equated  period  :  — 

217750 
9105 


9,105 


=  23-915,  or  24  years. 


217,750 


The  True  or  Mathematical  Method  of  Finding  the 
Equated  Period  of  Repayment.  As  already  stated,  the 
method  about  to  be  described  differs  slightly  from  that 
adopted  in  the  previous  example,  and  is  shorter.  The  first  step 
is  to  find  by  calculation  on  standard  calculation  form,  Xo.  2, 
the  present  values  of  the  component  parts  of  the  loan  for  the 
respective  periods  allowed.  These  present  values  are  shown  in 
the  sixth  column  in  Table  XXXII.  F.,  and  amount  together  to 
£4663'04.  The  next  step  is  to  find  the  number  of  years  in 
which  £4463-04  will  amount  to  the  loan  of  £9,105,  at  3  per 
cent.,  in  order  to  replace  a  gradual  repayment  of  the  component 
parts  of  the  loan  by  a  simultaneous  repayment  of  the  whole. 
In  the  previous  example,  three  methods  are  described  of  finding 
the  number  of  years,  one  being  by  direct  calculation  by  means 
of  the  formula  relating  to  the  amount  of  £1  per  annum  in 
standard  calculation  form,  Xo.  3,  which  is  illustrated  by 
Statement  XXXII.  E.  The  second  method  of  finding  the 
number  of  years  is  by  means  of  Thoman's  table-,  and  is  fully 
described  in  the  previous  example.  The  third  method  is  by 
trial  and  error,  based  upon  the  approximate-  ininiber  of  years 
in  the  equated  period  found  by  the  arithmetical  method, 
and  this  method  will  be  applied  to  the  present  instance.  T'sing 
standard  calculation  form  Xo.  1,  it  may  be  found  that  £4663'04, 
accumulated  at  3  per  cent.,  will  amount  to:  — 


THE  EQUATION  OF  THE  PERIOD  OF  REPAYMENT      405 

in  24  years £9479-00 

in  23  years £9202-90 

in  22  years £8934-87 


as  compared  with  the  equated  period  as  found  by 
the  arithmetical  method  :  — 

in  24  years £9105-00 


It  is  obvious  that  the  true  period  is  nearer  to  23  years  than 
to  22  years.  If  it  be  calculated  exactly  by  either  of  the 
methods  used  in  the  previous  example  it  will  be  found  to  be 
22-638  years;  and  therefore  23  years  should  be  adopted  in 
practice  instead  of  24  years  as  found  by  the  arithmetical 
method  above  described.  The  actual  difference  between  the 
periods  found  by  the  two  methods  is  1-277  years.  In  the 
previous  example  the  actual  difference  was  1-864  years,  but  in 
that  case  the  longer  repayment  period  was  assumed  to  be  45 
years,  whereas  in  the  present  instance  it  is  30  years  only.  There 
is  not  any  common  ratio  existing  between  the  original  and  the 
equated  periods,  or  between  the  two  equated  periods  as  found 
by  the  arithmetical  and  true  methods.  The  number  of  years 
in  the  equated  period  depends  upon  the  interaction  of  the 
component  parts  of  the  loan  and  the  respective  periods  pre- 
scribed for  their  repayment. 

Having  found  the  true  equated  period  in  the  above  manner 
the  enquiry  strictly  comes  to  an  end,  but  if  the  annual  instal- 
ment is  required  to  be  spread  equally  over  the  period  it  may 
be  found  in  the  usual  manner  on  standard  calculation  form 
No.  3x.  So  far  as  the  lender  is  concerned  this  is  quite  equit- 
able, but  having  regard  to  the  varying  life  of  the  assets  created 
out  of  the  loan  the  question  of  the  annual  charges  to  revenue 
or  rate  during  the  period  becomes  important  and  will  be  fully 
considered  in  the  following  chapters. 


Section  VIII. 

The  Equation  of  the  Incidence  of 
Taxation. 


409 


GHAPTEE    XXXIII. 

THE  EQUATIOX  OF  THE  IXCIDEXCE  OF  TAXATION. 

Comparison  of  the  total  annual  loan  charges  to  revenue  or 
rate,  before  and  after  the  equation  of  the  period  of 
repayment,  showing  the  unequal  incidence  of  taxation 
if  the  annual  instalment  and  interest  upon  the  total 
loan  be  spread  equally  over  the  equated  period. 


The  subject  of  enquiry  in  the  previous  chapter  is  the  correct 
method  of  finding  tlie  equated  date  of  repayment  of  several 
loans  repayable  at  varying-  dates,  and  the  result  of  such  enquiry 
is  to  show  that  the  generally  adopted  arithmetical  method  is 
wrong  in  principle  seeing  that  it  tends  to  prolong  the  period. 
Having  found  the  equated  date  the  next  step  is  to  ascertain 
the  annual  instalments  to  be  charged  to  revenue  or  rate  account 
during  the  equated  period  in  substitution  for  the  annual  instal- 
ments required  under  the  original  conditions  before  equation. 
The  present  practice  is  to  regard  the  matter  purely  from  the 
point  of  view  of  the   loan  holder  and  to  set   aside   an   equal 
annual  instalment  to  be  spread  over  the  Avhole  of  the  equated 
period  without  any  regard  to  the  incidence  of  taxation  or  the 
life  of  the  asset  created  out  of  the  loan.     Seeing,  however,  that 
the  annual  instalments  are  accumulated  in  the  sinking  fund 
and  are  not  repaid  to  the  lender  until  the  end  of  the  period  it 
is  immaterial  to  him  how  the  annual  instalments  are  distributed 
over  the  revenue  or  rate  accounts  of  the  equated  period.     On 
the  contrary  it  is  a  matter  of  concern  to  the  ratepayer  that  the 
annual  contributions  out  of  revenue  or  rate  are  borne  equitaldy 
by  successive  years,  and  this  question  will  now  be  considered. 
The  permanent  character  of  the  security  for  local  loans  is  shown 
by  the  preferential  nature  of  the  redemption  of  part  of  the 
loan  out  of  the  sinking  fund  before  maturity.     In  the  case  of 
financial  and  commercial  undertakings  any  such  redemptions 
are  made  pro  rata  or  by  some  method  in  which  each  loanholder 
has  an  equal  chance.  . 

The  effect  of  the  generally  adopted  method  of  equation  of 
the  period  of  repayment  is  to  reduce  the   annual   instalment 


4IO    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

during  the  earlier  years  of  the  equated  repayment  period  and 
thus  relieve  the  revenue  or  rate  account  of  those  years.  This 
may  at  hrst  sight  appear  strange,  when  it  is  borne  in  mind  that 
under  an  equation  of  the  period  the  total  loan  is  repaid  at  an 
earlier  date  although  the  mathematical  result  may  be  exactly 
equal. 

This  will  be  found  to  be  the  case  on  referring  to  Table 
XXXII.  D.  in  Chapter  XXXII,  giving  the  annual  instalments 
required  to  repay  a  loan  of  £56,000  in  an  equated  period  of 
2o  years  in  substitution  for  periods  of  5,  15,  29  and  45  years. 

The  following  table  (XXXIII.  A.)  shows  the  annual  instal- 
ments to  be  set  aside,  dividing  the  original  periods  into  five,  at 
the  end  of  four  of  which,  part  of  the  loan  would  have  been 
repaid,  whereas  the  total  loan  is  repayable  at  the  end  of  the 
23rd  year  under  the  equated  method  :  — 


TABLE  XXXIII.  A. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature 
having  prescribed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Comparison  of  the  annual  charges  to  revenue  or  rate  in  respect 
of  the  annual  instalments  under  (1)  the  original  conditions, 
and  (2)  after  equation  where  such  annual  instalments  are 
spread  equally  over  the  equated  period. 


(1)  Original 
periods. 

(•2)  Equated 
period. 

Equated  period 
as  coni])ared  with 
original  periods. 

Periods  of 
equal  incidence. 

No.  of 
years. 

Sinking  fund 
instalments. 

No.  of 
years. 

Sinking  fund 
instalments. 

Increase. 

Decrease. 

5  years 

5 

2217-25 

5 

1725-58 

— 

491-67 

10  years 

10 

1840-54 

10 

1725-58 

— 

114-96 

8  years 

8 

550-14 

8 

1725-58 

1175-44 

— 

6  years 

6 

550-14 

— 

— 

— 

55014 

16  years 

16 

107-85 

— 

— 

— 

107-85 

45 


23 


The  above  table  shows  that  during  the  first  five  years  of  the 
equated  period  the  annual  instalment  is  reduced  by  £491-67, 
and  that  during  the  second  period  of  10  years  there  is  a  similar 


THE  EQUATION   OF  THE  INCIDENCE  OF  TAXATION      411 

aunuai  reductiou  oi  <£li4  y6.  itie  heaviest  cliarge  falls  upon 
the  third  and  huai  period  of  eight  years,  which  is  part  of  an 
original  period  of  i4  years.  Uuring  tiiis  period  tlie  annual 
instalment  is  greater  by  i;il75"44  than  the  corresponding 
annual  instalment  under  the  original  conditions  before  equa- 
tion. This  large  increase  in  the  annual  instalment  is  due  to 
the  fact  that  under  the  original  conditions,  before  equation, 
£26,000  of  loan  would  have  been  repaid  by  the  end  of  the  15th 
year,  being  the  end  of  the  second  portion  both  of  the  equated 
and  original  periods.  This  amount  of  loan,  having  a  short 
period  of  repayment,  naturally  required  a  larger  annual  instal- 
ment than  the  remaining  loans  having  longer  periods  of  repay- 
ment. After  the  final  repayment  of  the  loan,  at  the  end  of  the 
equated  period  of  23  years,  by  means  of  the  equal  annual 
equated  instalment  of  £1725'58,  the  revenue  or  rate  account  is 
relieved  of  all  contributions  both  in  respect  of  the  annual 
instalment  and  interest  upon  the  loan. 

As  already  pointed  out,  the  actual  figures  in  individual  cases 
will  vary  in  accordance  with  the  amounts  of  the  respective 
loans  and  the  length  of  the  various  periods  allowed  for  repay- 
ment, but  the  generality  of  equations  will  follow  the  main 
features  here  outlined.  The  results  of  an  equation  of  the  period 
of  repayment  may  be  summarised  as  follows  :  — As  regards  the 
annual  sinking  fund  instalment  to  be  charged  to  revenue  or 
rate  account,  the  earlier  and  later  years  of  the  original  repay- 
ment period  will  be  relieved  and  the  resulting  burden  thrown 
upon  the  middle  portion  of  the  original  repayment  period, 
which  is  the  final  part  of  the  amended  equated  period.  This 
relief  will  of  course  apply  in  full  to  that  part  of  the  original 
repayment  period  beyond  the  equated  period,  seeing  that  the 
whole  of  the  loan  will  then  have  been  repaid. 

The  following  table,  XXXIII.  B.,  shows  the  result  of  the 
equation  of  the  period  of  repayment,  as  regards  the  interest 
upon  the  loan,  chargeable  against  the  revenue  or  rate  account 
of  each  year  of  the  equated  period  as  compared  with  the  corres- 
ponding annual  interest  charges  under  the  original  conditions, 
before  equation.  Under  the  original  conditions  the  loan  would 
have  been  gradually  repaid,  thereby  reducing  the  annual 
interest  charges  against  the  revenue  or  rate  accounts  of  subse- 
quent years,  but  after  the  equation  of  the  period  of  repayment, 
the  revenue  or  rate  account  of  each  year  of  the  equated  period 
is  charged  with  interest  upon  the  total  amount  of  the  loan. 
The  equation  of  the  annual  charge  for  interest  upon  the  loan 
will  be  fully  considered  in  Chapter  XXXV. 


412         REPAYMENT   OF   LOCAL   AND   OTHER   LOANvS 


TABLE  XXXIII.  B. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature 
having  prescribed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Comparison  of  the  annual  charges  to  revenue  or  rate  in  respect 
of  interest  upon  the  loan  under  (1)  the  original  conditions, 
and  (2)  after  equation. 


Periods  of  equal 
incidence. 

(1)  Original 
periods. 

Loan.             Interest. 

(2)  Equated 
period. 

Loan.              Interest. 

Equated  period 
as  compared  with 
original  periods. 

Increase.       Decrease 

5  years 

56000 

1960 

56000 

1960 



10  years 

54000 

1890 

56000 

1960 

70         — 

8  years 

;  50000 

1050 

56000 

1960 

910          — 

6  years 

;{0000 

1050 

— 

— 

—        1050 

16  years 

10000 

350 

— 

— 

—          350 

45  years 

The  foregoing  tables  show  that  under  the  generally  adopted 
equated  method  the  relief  during  the  first  period  of  5  years  is 
solely  .in  respect  of  the  annual  sinking  fund  instalment  and 
that  the  annual  interest  charges  are  unaltered  owing  to  the 
fact  that  no  part  of  the  loan  is  repayable,  under  the  original 
conditions,  until  the  end  of  the  fifth  year.  During  the  second 
period  of  10  years  there  is  a  decrease  in  the  annual  instalment, 
but  there  is  an  increase  in  the  amount  of  the  annual  interest 
charges,  because  the  repayment  of  £2000  of  loan  which,  imder 
the  original  conditions,  would  have  been  made  at  the  end  of 
the  fifth  year,  has  by  the  equation  of  the  period  of  repayment 
been  deferred  until  the  end  of  the  23rd  year.  The  third  period 
of  (S  years,  being  the  final  portion  of  the  equated  period,  has  to 
bear  an  increased  annual  charge  of  £2085' 44,  being  an  increase 
in  the  annual  instalment  of  £1175'44  in  addition  to  an 
increased  annual  interest  charge  of  £910. 

The  explanation  of  the  large  increase  in  the  total  annual 
burden  imposed  upon  this  period  is,  that  it  has  to  bear,  not  only 
the  relief  to  the  earlier  periods  of  5  and  10  years,  but  also  tlie 
total  relief  to  that  part  of  the  original  repayment  period  which 
is  beyond  the  equated  period.  The  whole  of  the  foregoing 
conclusions  are  shown  in  the  following  table :  — 


THE  EQUATION  OF  THE  INCIDENCE  OF  TAXATION      413 

TABLE  XXXIII.  C. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature, 
liaving-  prescribed  periods  of  repayment)  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Showing  the  variation  in  the  total  annual  charges  to  revenue 
or  rate  in  respect  of  the  sinking  fund  instalment  and 
interest  upon  the  loan,  under  (1)  the  original  conditions 
and  (2)  after  equation,  where  such  annual  instalments  are 
spread  equally  over  such  equated  periods. 

A  Summary  of  Tables  A  and  B  above. 


Periods  of  equal 
incidence. 

5  years 
10  years 

8  years 

6  years 
16  years 

45  years 


Sinking  fund 
instalment. 


Annual  interest 
on  loan. 


Total  charge  to 
revenue  or  rate. 


Increase.  Decrease.  Increase.  Decrease.      Increase.  Decrease. 

—  491-67         _          _           —  491-67 

—  114-96      7000       —           —  44-96 
1175-44       ^  91000       —      2085-44  — 

—  55014       —  1050-00       —  160014 

—  107-85       —  350-00       —  457-85 


These  results  are  so  remarkable  that  some  enquiry  may 
profitably  be  made  into  the  matter,  not  only  as  to  the  necessity 
to  make  the  equation,  but  also  as  to  the  effect  of  the  equation 
upon  the  incidence  of  taxation.  The  necessity  to  make  an 
equation  of  the  period  may  arise  in  several  ways.  The  most 
important  is  in  order  to  provide  for  the  repayment  of  a  loan 
raised  by  the  issue  of  stock  redeemable  on  a  fixed  date  where 
the  loan  is  authorised  for  works  having  varying  lives  or  periods 
of  utility.  In  the  case  of  a  loan  to  provide  for  outlay  of  one 
character  only,  or  for  different  classes  of  outlay  having  the  same 
life  or  duration  of  utility,  there  is  not  any  necessity  to  make 
an  equation,  the  calculation  being  a  simple  one.  The  governing 
factor  is  the  unequal  life  or  duration  of  utility  of  the  works 
authorised,  upon  which  are  based  the  periods  allowed  for  tlic 
repayment  of  the  component  parts  of  the  loan. 

A  further  need  for  the  equation  of  the  period  of  repaymeut 
arises  on  the  consolidation  of  several  loans  repayable  at  various 
dates.     This  may  be  part  of  a  large  financial  scheme  undertaken 


414  REPAYMENT   OF    LOCAL   AND   OTHER   LOANvS 

with  the  object  of  generally  simplifying  the  finances  of  a  local 
authority  or  on  the  issue  of  stock  to  replace  a  number  of  small 
loans  borrowed  for  short  periods.  The  issue  of  such  a  stock 
avoids  the  necessity  of  reborrowing  a  large  number  of  small 
sums  continually  falling  due,  and  gives  a  permanency  to  the 
outstanding  debt.  It  also  considerably  simplifies  the  sinking 
fund  book-keeping  and  renders  much  easier  not  only  the 
investment  of  the  sinking  fund  but  also  the  redemption  of  part 
of  the  loan  during  the  operation  of  the  fund.  Further,  investors 
prefer  a  stock  of  large  amount  which  is  quoted  in  the  Stock 
Exchange  list  and  is  readily  saleable.  Looking  at  the  matter 
from  the  investor's  point  of  view,  it  is  a  coincidence  perhaps 
that  the  equated  period  generally  found  necessary  is  about 
20 — 30  3^ears,  which  is  as  long  as  investors  generally  approve. 
Both  the  shorter  and  longer  repayment  periods  allowed  for  the 
repayment  of  local  debt  are  not  suitable  for  permanent  invest- 
ment as  a  stock,  and  local  authorities  are  thus  obliged  to  rely 
upon  the  small  investor  who  causes  much  more  administrative 
work  than  the  holders  of  a  stock.  The  renewal  or  reborrowing 
of  small  loans  falls  upon  the  officials  of  the  local  authority, 
whereas  the  burden  of  any  change  in  the  ownership  of  a 
registered  stock  is  borne  by  the  holder  except  the  registration 
of  the  transfer  and  the  preparation  of  the  new  certificate. 

The  investor  may  therefore  be  eliminated  from  the  enquiry 
because  if  he  is  willing  to  accept  payment  on  the  equated  date 
the  arithmetical  or  mathematical  methods  of  ascertaining  that 
date  both  give  a  sufficiently  near  approximation.  The 
investor,  except  in  a  very  academic  way,  is  not  concerned  with 
the  annual  charges  for  redemption  of  the  loan.  The  effect  of 
the  equation  of  the  period  upon  the  incidence  of  taxation 
therefore  becomes  the  principal  subject  of  enquiry,  and  the 
above  table  (XXXIII.  C.)  shows  that  there  is  a  very  wide  differ- 
ence as  between  the  original  and  the  equated  periods  in  regard 
to  the  burden  imposed  upon  successive  years  or  periods  of  years. 
It  is  here  advisable  to  recapitulate  the  principles  governing  the 
method  of  fixing  the  original  periods  of  repayment.  The 
predominant  factor  in  fixing  the  proper  repayment  periods  to 
be  alloM'ed  in  respect  of  each  individual  class  of  outlay  is  found 
in  the  principle  laid  down  in  the  Statutes  and  adopted  in  the 
practice  of  Parliament  and  the  Government  departments, 
namely,  that  all  loans  shall  be  repaid  during  the  period  of 
utility  or  duration  of  the  works  in  respect  of  which  ihe  loan  was 
borrowed.  But  a  local  authority  has  not  any  capital  and  can 
only  repay  the  loan  by  annual  contributions  out  of  rate  or  out 


THE  EQUATION  OF  THE  INCIDENCE  OF  TAXATION      415 

of  the  profits  of  its  revenue  earning  undertakings.  It  may  be 
contended  that  revenue  earning  undertakings  should  be  treated 
in  a  different  manner  to  purely  spending  departments,  such  as  a 
sanitary,  highway  or  education  authority,  where  the  annual 
expenditure,  both  for  current  expenses  and  debt  redemption 
charges,  is  taken  direct  from  the  pockets  of  the  community  by 
way  of  a  rate.  Here  it  is  important  to  adjust  the  incidence  of 
taxation  very  accurately — and  this  is  the  object  of  the  careful 
scrutiny  by  Parliament  and  the  Government  departments  of 
the  periods  of  repayment  allowed.  The  effect  of  this  scrutiny 
is  seen  in  the  original  repayment  periods  allowed  which  are 
generally  fixed  at  a  number  of  years  well  within  the  life  of  the 
works  for  which  the  loan  is  authorised.  If  these  periods  are 
properly  allowed  and  the  sinking  fund  instalments  are  based 
upon  them  there  is  an  equitable  incidence  of  the  annual  burden, 
and  the  annual  instalments  may  then  properly  be  considered  as 
the  equivalent  of  an  annual  charge  for  depreciation — thereby 
carrying  out  the  principle  laid  down  in  an  earlier  chapter  of 
making  each  ratepayer  contribute  annually  his  due  proportion 
of  the  cost  of  the  benefits  he  receives  each  year,  whether  that 
cost  be  paid  for  during  the  year  or  be  spread  over  a  series  of 
years. 

But  the  equation  of  the  period  of  repayment  is  purely  a 
financial  operation,  and  relates  solely  to  the  date  of  repayment 
of  the  loan  without  any  regard  to  the  effect  of  such  equation 
upon  the  annual  charges  to  the  community  by  way  of  rate. 
The  case  is  different  with  a  commercial  or  financial  undertaking 
where  the  repayments  of  debt  are  made  out  of  the  general  assets 
of  the  concern  and  are  not  charged  against  the  profit  and  loss 
account  except  and  in  so  far  as  the  operations  of  each  individual 
year  cause  a  loss  of  capital  due  to  wear  and  tear  of  the  asset. 
The  repayment  of  debt  and  the  annual  charge  to  revenue  are 
in  the  case  of  such  undertakings  kept  severely  separate  and 
distinct.  In  the  case  of  a  local  authority  the  conditions  are 
the  exact  opposite.  In  the  first  place,  there  is  a  careful  and 
searching  enquiry  by  Parliament  and  the  Government  depart- 
ments, with  the  object  of  fixing  the  annual  amounts  to  be 
charged  to  the  revenue  or  rate  accounts  of  successive  years  in 
respect  of  the  repayment  of  the  debt  and  the  consequent  charges 
for  interest.  These  total  charges  are  in  many  cases  regularly 
met  out  of  revenue  or  rate  during  a  part  of  the  period  so 
allowed,  and  it  may  then  become  necessary  or  advisable  to 
make  an  alteration  in  the  date  at  which  the  loan  shall  be 
repaid,  and  an  equation  of  the  period  of  repayment  is  made 


4i6         REPAYMENT    OF    LOCAL   AND    OTHER   LOANvS 

resulting  in  such  a  drastic  rearrangement  of  tlie  total  annual 
charges  that  the  original  careful  calculations  as  to  the  life  of 
the  asset  are  ignored  and  rendered  valueless.  Reverting  to  the 
present  example,  it  will  be  seen  from  Table  XXXIII.  C.  that, 
although,  in  consequence  of  the  equation  of  the  period,  the  final 
repayment  of  the  loan  is  expedited,  there  is  actually  a  decrease 
in  the  annual  burden  for  the  next  15  years  and  an  absolute 
relief  from  any  burden  whatever  during  the  later  years  of  the 
original  period  which  were,  under  the  conditions  existing  at  the 
time  the  loan  was  authorised,  charged  with  their  due  proportion. 
And  the  whole  of  the  added  burden  is  imposed  upon  the  final 
years  of  the  newly  ascertained  or  equated  period  at  a  time 
when  probably  the  undertaking  may  have  to  incur  outlay  on 
renewals. 

If  it  at  any  time  becomes  necessary  or  advisable  to  expedite 

the   repayment   of   the   loans   the   calculation    of   the   equated 

period,  and  the  resulting  amended  annual  instalment  should  be 

made  in  such  a  manner  as  to  impose  a  proportionate  part  of  the 

additional   burden  upon   each  year   of  the  equated   period   of 

repayment,  instead  of,  as  is  the  present  practice,  relieving  both 

the  earlier  and  the  later  years  of  the  original  repayment  period 

at  the  expense  of  the  middle  portion  of  that  period  which  is  also 

the  final  portion  of  the  equated  period.     A  method  of  doing  this 

as  regards  the  annual  instalment  will  be  described  in  Chapter 

XXXIY,  and  as  regards  the  interest  upon  the  loan  in  Chapter 

XXXY.    Fp  to  this  point  the  general  question  of  the  repayment 

of  debt  has  been  treated  from  an   actuarial  or  mathematical 

standpoint  only,  but  the  disturbing  element  now  introduced  by 

the  necessity  to  accelerate  the  final  repayment  of  the  loan  and 

to  vary  the   dates   of  repayment,    by   substituting   therefor   a 

common  'date   for   the   repayment    of   the   whole   of   the  loan, 

depends  upon  circumstances  which  are  generally  of  a  variable 

and  somewhat  accidental  nature.     It  is  therefore  necessary  to 

find   an  equitable  practical   method   of  restoring   the   original 

status,  namely,  to  charge  the  revenue  or  rate  account  of  each 

year  with  its  due  proportion  of  the  annual  burden  of  redemption 

and  interest   charges.     At  this  stage   it  becomes   advisable  to 

differentiate   between   the    annual    charges    in    respect    of   the 

redemption   of  debt   and  the  interest  payable   upon   the  loan, 

because  where  loans  are  repaid  by  means  of  a  sinking  fund. 

interest  is  payable  upon  the  total  amount  of  the  loan  during  the 

whole    of   the    redemption    period    of    whatever  duration,    and 

ceases   entirely    at    the    end    of   that    ])eiiod.      Any    reduction 

therefore    in    the    period    of    redemption   will    corrosjinndingly 


THE  EQUATION  OF  THE   INCIDENCE  OF   TAXATION      417 

reduce  the  period  during  which  there  is  any  charge  whatever 
in  respect  of  interest  upon  the  loan,  and  it  will,  in  the  first 
instance,  be  assumed  that  it  is  perfectly  equitable  to  ignore  the 
relief  of  later  years  in  respect  of  interest  upon  the  loan  due  to 
an  increased  sinking  fund  burden  imposed  upon  the  earlier 
years  in  consequence  of  the  accelerated  final  extinction  of  the 
debt.  Confining  the  enquiry  therefore  solely  to  the  sinking  fund 
instalment,  annually  charged  to  the  revenue  or  rate  account,  it 
is  necessary  to  revert  to  the  primary  factor  in  the  redemption 
of  debt,  namely,  the  life  or  duration  of  continuing  utility  of  the 
asset  created  out  of  the  loan.  A  broad  line  is  here  required  to 
be  drawn  between  the  two  objects  of  the  annual  instalment, 
namely,  the  repayment  of  the  debt  and  the  charge  to  revenue  or 
rate  of  the  cost  of  the  asset  during  its  life  or  period  of  utility. 

Under  the  original  conditions  before  equation  the  total  debt 
would  have  been  gradually  repaid  at  the  end  of  a  series  of 
periods  up  to  the  45th  year,  whereas  under  the  amended 
conditions  the  total  loan  will  be  repaid  in  one  sum  at  the  end 
of  the  23rd  year  or  approximately  in  about  one  half  the 
number  of  years.  The  magnitude  of  the  annual  instalment  to 
be  set  aside  and  charged  to  revenue  or  rate  in  order  to  redeem  a 
given  loan  depends  primarily  upon  the  period  allowed  for  its 
redemption,  which  is  based  upon  the  life  of  the  asset;  and 
therefore  if  the  total  period  be  reduced,  the  periods  allowed  for 
the  repayment  of  the  component  parts  of  the  loan  should  be 
correspondingly  reduced. 


A  c 


4i8         REPAYMENT   OF   LOCAL   AND    OTHER   LOANvS 


CHAPTER  XXXIV. 

THE  EQUATION  OF  THE  INCIDENCE  OF  TAXATION 

{Continued.) 

THE  ANNUAL  INSTALMENT. 

The  various  methods  of  adjusting  the  annual  charges  to 
revenue  or  rate  during  the  equated  period  in  propor- 
tion to  the  life  or  duration  of  continuing  utility  of 

the  asset  created  out  of  the  loan,  viz  : by  charging 

the  revenue  or  rate  account  of  each  year  of  the 
equated  period  with  the  annual  instalment  chargeable 
against  each  year,  before  equation.  and  in  addition 
thereto  a  supplementary  annual  instalment  : 

[a]  to  be  spread  equally  over  the  equated  period,  or 

(6)  to  be  proportionate,  year  by  year,  to  the  annual 
instalments  before  equation. 


The  previous  argument  will  now  be  applied  to  tlie  example 
under  review,  namely,  tlie  repayment  of  a  loan  of  £56,000, 
authorised  for  large  public  works,  the  component  parts  of  wliicli 
have  varying  periods  of  continuing  utility  and  consequent 
prescribed  periods  of  repayment.  Under  tlie  original  conditions 
the  repayment  of  the  loan  Avas  spread  over  a  period  of  45  years, 
but  under  the  altered  conditions  it  is  required  that  the  whole 
of  the  loan  shall  be  repaid  at  the  end  of  an  equated  period  of 
23  years.  In  this  chapter  the  correct  method  of  spreading  the 
actual  burden  equitably  over  the  equated  period  is  the  subject 
of  enquiry  and  not  the  method  of  finding  the  true  equated 
period  which  has  been  fully  discussed  in  Chapter  XXXII. 

The  object  is  to  distribute  the  annual  sinking  fund  burden 
equitably  over  the  rcHluced  period  of  repnyment  instend  of 
imposing  an  undue  burden  upon  the  final  yeors  of  the  equated 
period  which  is  the  effect  of  the  mefliod  generally  adopted,  as 
shown  in  Table  XXXIII.  C. 

For  this  purpose  it  Avill  be  an  advantage  to  tal)u]ate  the 
original    conditions    in    the    example    considered    in    Chapter 


THE    INCIDENCE    OF    TAXATION  419 

XXXII,  and  to  show  also  tlie  proportionate  amended  periods  of 
repayment  under  the  amended  conditions  due  to  the  equation 
of,  or  alteration  in,  the  final  period  of  repayment,  as  follows  :  — 

TABLE  XXXIV.  A. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature, 
having  prescribed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Comparison  of  the  annual  charges  to  revenue  or  rate  in  respect 
of  the  annual  instalments  under  (1)  the  original  conditions, 
and  (2)  after  equation  if  such  instalments  are  increased 
in  proportion  to  the  reduction  in  the  repayment  period. 

Rate  of  accumulation,  3  per  cent. 

Number  of  years  allowed  Annual  sinking  fund 

for  repayment  of  loan.  instalments.     3  per  cent. 

Amount  Equated  23  years 

Class  of  of  loan  Original  period  of  45  year  equated 

outlay.  authorised,  conditions.         23  years.  period.  period. 

A  10,000  45  23  107-85  308-14 

B  20,000  29  14  442-29  1170-53 

C  24,000  15  8  1290-40  2698-95 

D  2,000  5  3  376-71  64706 

56,000  2217-25  4824-68 


The  above  annual  instalments  under  the  23  year  equated 
period  will  repay  the  component  parts  of  the  loan  at  the  end 
of  the  respective  reduced  periods  of  repayment,  and  are  not  in 
any  way  equated  instalments.  They  are  placed  here  only  to 
show  the  effect  of  reducing  the  period  from  45  to  23  years. 
They  illustrate  forcibly  the  wide  difference  between  the  financial 
obligation  to  repay  the  loan  and  the  anticipated  life  of  the 
various  classes  of  outlay.  Both  these  factors  are  distinct,  but 
the  generally  adopted  method  of  fixing  the  future  annual 
instalment  after  ascertaining  the  equated  period  does  not  make 
any  such  distinction  but  treats  them  as  being  equivalent. 

It  may  be  at  once  stated  that  it  is  not  possible  to  adjust  the 
unequal  burden  of  the  annual  instalment  as  ascertained  by  the 
generally  adopted  method,  by  reducing  each  of  the  periods 
allowed  for  the  component  parts  of  the  outlay  in  proportion  to 
the  reduction  in  the  final  period  of  repayment  as  shown  in  the 
above  table.  There  are  so  many  disturbing  factors  and  the 
conditions   are   so  widely  altered  by  reducing  the  periods  by 


420         REPAYMENT    OF    LOCAL   AND    OTHER   LOANvS 

one  half  that  the  results  obtained  show  wider  differences  than 
exist  under  the  method  now  in  use.  The  author  has  worked  out 
the  problem  in  detail,  but  the  results  are  too  long  to  give  in 
full  and  would  not  be  of  any  practical  A^alue. 

It  is  therefore  necessary  to  adopt  another  line  of  enquiry 
in  order  to  find  a  method  of  determining  the  annual  instalment 
or  instalments  to  be  charged  to  revenue  or  rate  to  repay  the 
total  loan  in  one  sum  at  the  end  of  the  e(|uated  period.  So  far 
as  the  investor  is  concerned  the  equated  period  of  repayment 
already  found  is  comparatively  correct,  but  a  more  important 
matter  is  the  equated  annual  incidence  of  the  burden  upon  each 
year's  revenue  or  rate  account.  So  long  as  the  period  of  repay- 
ment and  the  life  of  the  asset  are  the  same  the  two  methods 
yield  identical  results,  but  any  alteration  in  the  term  of  repay- 
ment makes  an  important  difference  in  the  annual  charges  to 
revenue  or  rate  in  successive  years.  By  the  method  now 
generally  adopted  and  previously  described  the  annual  instal- 
ment is  spread  over  the  whole  of  the  equated  period  of  repay- 
ment, and  this  is  considered  to  be  an  equitable  substitute  for  a 
gradually  decreasing  charge  which  has  been  ascertained  after 
careful  enquiry  as  to  the  life  of  the  asset. 

Very  little  consideration  will  show  that  this  is  very  far  from 
being  correct;  and  the  result  of  the  previous  enquiry  is  to  show 
that  although  the  effect  of  the  equation  is  to  reduce  the  period 
of  repayment  and  correspondingly  increase  the  total  charge  for 
redemption,  yet  there  is  actually  a  reduction  in  such  annual 
charge  during  the  earlier  years  of  the  equated  period  and  an 
absolute  relief  from  any  charge  whatever  during  that  part  of 
the  original  period  beyond  the  equated  period.  These  reduc- 
tions in  the  charsres  against  the  earlier  and  later  vears  of  the 
original  period  involve  a  severe  additional  annual  burden  upon 
the  later  years  of  the  equated  period.  Although  this  inequality 
is  not  fully  appreciated  yet  its  effect  has  been  mentioned  in 
several  places  in  the  reports  of  the  parliamentary  committees 
which  have  enquired  into  the  finances  of  local  authorities  where 
it  is  pointed  out  that  under  an  equated  method,  loans  for  out- 
lays for  which  short  terms  are  generally  allowed  are  not  repaid 
until  the  end  of  the  longer  equated  period,  and  consequently 
further  borrowing  powers  ought  not  to  be  granted  when  the 
asset  is   exhausted. 

The  result  of  the  previo\is  discussion  of  the  subject  is  to 
em])hasise  the  fact  that  as  regards  the  annual  charge  to  revenue 
or  rate  the  most  important  factor  is  the  life  of  the  asset,  and  it 
may  naturally  be  concluded  that  if  any  change  is  made  in  the 


THE    INCIDENCE    OF    TAXATION  421 

final  period  of  repayment  the  amended  annual  instalment  to  be 
charged  to  revenue  or  rate  account  should  continue  to  bear  as 
near  as  possible  an  approximate  ratio  to  the  original  charge, 
instead  of,  as  is  the  present  practice,  spreading  the  burden 
equally  over  the  equated  period  without  any  regard  to  the  life 
of  the  asset. 

A  method  of  doing  this  will  now  be  fully  described,  taking 
as  an  example  the  loan  of  £56,000  already  used  to  illustrate  the 
previous  remarks  in  Chapter  XXXII  upon  the  method  of 
finding  the  true  equated  period.  In  comparing  the  two 
methods  it  is  important  to  bear  in  mind  that  where  the  periods 
are  not  equated  the  several  sinking  funds  will  mature  at 
successive  dates,  at  each  of  which  portions  of  the  original  loan 
will  be  repaid  out  of  such  funds,  whereas  under  the  equated 
method  the  whole  loan  is  repayable  on  one  date.  Further,  the 
equated  period  covers  the  date  of  final  repayment  of  one  or  more 
of  the  component  parts  of  the  original  loan.  In  the  present 
chapter  the  sinking  fund  instalment  only  will  be  considered, 
leaving  out  of  account  for  the  moment  the  interest  payable 
upon  the  loan  which,  as  already  pointed  out,  ceases  entirely  at 
an  earlier  date  under  the  equated  method,  although  the  later 
years  of  the  equated  period  bear  a  larger  interest  charge  than 
they  do  under  the  original  conditions.  This  is  shown  by 
Table  XXXIII.  B. 

The  following  method  of  adjusting  the  annual  charge  to 
revenue  or  rate  after  an  equation  of  the  period  is  based  upon 
the  Relative  periods  allowed  for  the  repayment  of  the  component 
parts  of  the  loan,  as  expressed  by  the  sinking  fund  instalments 
originally  found  requisite  to  repay  the  several  portions  of  the 
loan  at  the  end  of  the  respective  periods  prescribed.  In  order 
to  make  the  adjustment  it  is  first  requisite  to  ascertain  the 
amount  which  would  have  been  in  the  fund  if  the  original 
instalments  had  been  allowed  to  accumulate  until  the  end  of 
the  equated  period  of  23  years  instead  of  repaying  £2,000  at 
the  end  of  the  5th  year  and  a  further  £24,000  at  the  end  of  the 
15th  year.     The  original  annual  instalments  are  as  follows  :  — 

for  the  first  five  years     £2217-25  per  annum. 

for  the  next  ten  years 1840-54 

for  the  final  eight  years 550-14         ,, 

and  the  following  table  shows  the  amount  which  would  be  m 
the  fund  at  the  end  of  the  2ord  year  under  the  above  conditions. 


422  REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 

STATEMENT  XXXIV.  B. 

Loan  of  £56,000  (as  above). 

Sliowing  tlie  amount  which,  will  be  in  the  sinking  fund  at  the 
end  of  the  equated  period  of  23  years  if  the  original  annual 
instalments  as  shown  in  Table  XXXIII.  A.,  are  set  aside 
for  the  periods  originally  prescribed  and  no  part  of  the 
fund  is  applied  in  repaying  the  loan. 

Interest  at  3  per  cent. 

1.  Amount  of  iJ2217'25  per   annum  for  5 

years     

Amount  thereof  at  the  end  of  a  further 
18  years       

2.  Amount  of  £1840"54  per  annum  for  10 

years     

Amount  thereof  at  the  end  of  a  further 
8  years         

3.  Amount  of  £550'14  j)er  annum  for  8  years 


11772 


21100 


20040 


26728 
4892 


Total  amount  in  the  fund  at  end  of  23rd  year     51660 


being  :  — 

Loan  repayable  at  end  of  5th  year 

accretions  for  18  years     

Loan  repayable  at  end  of  15th  year     ... 
accretions   for  8  years     

Annual  instalment  of  £550"  14  accumu- 
lated for  23  years     

Amount    as    above     


2000 
1404 

24000 
6403 


3404 

30403 
17853 
51660 


The  above  amount  of  £51,660  which  would  have  been  iu  the 
fund  if  the  original  annual  instalments  had  been  added  thereto 
and  allowed  to  accumulate  for  23  years,  represents  that  portion 
of  the  loan  of  £56,000  which  would  have  been  repaid  by  means 
of  the  annual  charges  to  the  revenue  or  rate  accounts  of  the 
23  years.  These  annual  instalments  are  based  upon  the 
respective  repayment  periods  proper  to  be  allowed  for  the 
component  parts  of  the  outlay  and  may  be  accepted  as  fair  and 
])r()per  charges  against  each  year's  revenue  or  rate,  irrespective 
of  the  equated  period  allowed  for  tlie  repayment  of  the  loan. 


THE    INCIDENCE    OF    TAXATION  423 

The    balance    of    loau    thus    unprovided    for    is    arrived    at    as 
follows  :  — 

Total  amount  of  loan     £56000 

Amount  which  will  be  provided  by  the  accumula- 
tion of  the  original  annual  instalments  for 
23  years      £51660 


Amount  which  will  be  unprovided  for £4340 


This  amount  represents  the  deficiency  stated  in  terms  of 
the  loan,  by  which  the  accumulated  amount  of  the  original 
annual  instalments,  based  upon  the  life  of  the  asset,  will  be 
insufficient  to  repay  the  loan  at  the  end  of  the  amended  or 
equated  period,  and  this  amount  has  to  be  provided  by  a 
supplementary  instalment  or  instalments  to  be  set  aside  in 
some  way  during  the  equated  period  of  23  years.  To  be  per- 
fectly accurate,  the  annual  instalments  should  bear  the  same 
ratio  one  to  another  as  the  original  annual  instalments,  and 
this  should  be  done  where  the  amounts  of  loan  in  question  are 
large,  seeing  that  the  greater  portion  of  the  original  burden 
should  be  borne  by  the  earlier  years.  This  will  be  referred  to 
again  in  a  later  part  of  this  chapter  where  the  proper  mathe- 
matical method  of  making  the  adjustment  will  be  fully 
described.  For  the  present  it  will  be  assumed  that  it  is 
equitable  to  distribute  the  supplementary  annual  instalment 
equally  over  the  equated  period  of  23  years,  and  the  problem 
therefore  is  to  find  the  annual  sinking  fund  instalment  to  repay 
a  loan  of  £4340  in  23  years  at  3  per  cent.  This  supplementary 
instalment,  as  may  be  found  by  standard  calculation  form  3x, 
is  £133'73  per  annum,  and  the  ultimate  annual  instalments  to 
be  added  to  the  fund  during  the  successive  portions  of  the  23 
years  are  obtained  by  increasing  each  of  the  original  annual 
instalments,  prior  to  equation,  by  this  amount,  as  shown  in  the 
following   table  :  — 


424  REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 

TABLE  XXXIV.  C. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature, 
having  prescribed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Comparison  of  the  annual  charges  to  revenue  or  rate  in  respect 
of  the  annual  instalments  under  (1)  the  equated  method 
generally  adopted,  and  (2)  in  which  the  annual  instalments 
as  originally  ascertained  are  supplemented  by  an  equal 
additional  instalment  spread  over  the  equated  period. 


'eriods  of  equal 
incidence. 

Original 
annual 

instalments. 

Additional 

annual 
instalment. 

Total 

annual 

instalments. 

Annual 
instalment 
as  equated. 

5  years 

2217-25 

133-73 

2350-98 

1725-58 

10  years 

1840-54 

133-73 

1974-27 

1725-58 

8  years 

550-14 

133-73 

683-87 

1725-58 

23  years 

By  the  aid  of  the  above  figures  a  comparison  will  now  be 
made  between  the  annual  instalments  obtained  by  the  three 
methods,   namely :  — 

(1)  Instalments    payable     during     the    original     prescribed 

periods,  providing  for  the  gradual  repayment  of  the  loan 
at  the  end  of  5,  15,  29,  and  45  years,  and  which  are  based 
upon  the  life  of  the  asset. 

(2)  Instalments    payable    during    the    equated    period    only, 

based  upon  an  equal  annual  charge  to  the  revenue  or  rate 
account  of  each  year,  according  to  the  method  generally 
adoj)ted.  In  this  case  the  life  of  the  asset  is  not  taken 
into  account  in  fixing  the  annual  burden.  It  is  true 
that  the  periods  originally  prescribed  enter  into  the 
arithmetical  calculation  of  the  equated  period,  but  the 
effect  of  this  is  lost  by  spreading  the  instalment  equally 
over  the  whole  of  the  equated  period  so  ascertained. 

(3)  Instalments  payable  during  the  equated  period  only,  but 

which  are  not  equal  throughout  the  period  but  are  based 
upon  the  life  of  the  asset  and  are  approximately  pro- 
portionate to  the  instalments  before  equation. 

In  the  following  table  the  instalments  (2)  and  (3)  are 
compared  with  (1)  those  found  requisite  under  the  original 
conditions,  and  the  increase  or  decrease  in  the  annual  charge 
is  shown  in  respect  of  each  period  of  equal  incidence.  It  will 
be  noticed  that  the  final  8  years  of  the  equated  period  alone 
bear  any  increased  charge. 


THE    INCIDENCE    OF    TAXATION 


425 


TABLE  XXXIV.  D. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature, 
having  prescribed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Comparison  of  the  annual  charges  to  revenue  or  rate  in  respect 
of  the  annual  instalments  :  — 

1.  Based  upon  the  original  repayment  periods. 

2.  The  equated  method  generally  adopted. 

3.  The  annual  instalments,  as  in  Table  XXXIV.  C. 
The  following  increased  or  decreased  annual  charges  are  com- 
pared with  the  amounts  in  Column   1, 


Original 
periods. 

Periods  of  equal        Annual 
Incidence.         instalment. 


Equated  nietliod  usually 
adopted. 


Equated  metliod  previously 
described. 


5  years 
10  years 

8  years 

6  years 
16  years 


2217-25 

1840-54 
550-14 
550-14 
107-85 


Annual 
instalment. 

1725-58 
1725-58 
1725-58 


Increase+ 
Decrease— 


-491-67 
-114-96 
+  1175-44 
-55014 
-107-85 


Annual 
instalment. 

2350-98 
1974-27 

683-87 


Increase+ 
Decrease  — 

+  133-73 
+  133-73 
+ 133-73 
-550-14 
-107-85 


45  years 

The  above  table  shows  that  the  method  just  described, 
although  it  concerns  the  annual  instalment  only,  removes  the 
gross  inequality  which  exists  in  the  method  generally  adopted, 
in  which  each  year  of  the  third  period  of  8  years  is  charged 
with  £1 175-44  per  annum  more  than  is  the  case  under  the 
original  conditions  before  equation,  based  upon  the  life  of  the 
asset.  In  the  method  above  described,  not  only  is  there  a 
decreasing  annual  charge  due  to  the  fact  that  the  classes  of 
outlay  with  shorter  periods  of  utility  are  written  off  in  the 
earlier  years  but  the  total  relief  to  the  final  22  years  of  the 
original  period  (or  the  post  equated  period)  is  charged  against 
the  whole  of  the  equated  period.  This  is  almost  as  near  an 
equalisation  of  the  original  annual  burden  as  can  be  made,  and 
removes  the  objection  that  under  the  generally  adopted  method 
of  equation  the  redemption  of  loans  authorised  for  outlay  in 
respect  of  which  only  short  periods  are  granted  is  unduly 
delayed.  With  regard  to  the  annual  charges  in  respect  of 
interest  upon  the  loan,  there  is  not  any  variation  from  the 
conditions    previously    shown    under    the    generally    adopted 


426    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

method  of  equation,  and  Table  XXXIII.  B.,  showing  the  com- 
parison will  still  apply. 

At  this  point  it  is  interesting  to  compare  the  total  annual 
charges  for  sinking  fund  instalment  and  interest  upon  the  loan 
by  means  of  the  following  table  which  may  usefully  be  com- 
pared with  Table  XXXIil.  C. :  — 

TABLE  XXXIV.  E. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature 
having  prescribed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Showing  the  variations  in  the  total  annual  charges  to  revenue 
or  rate  under  the  method  described  in  Table  XXXIV.  C, 
as  compared  with  the  original  annual  instalments  before 
equation. 

This  table  should  be  compared  with  Table  XXXIIL  C. 


Periods  of  equal 
incidence. 

Sinking  fund 
instalment. 

Increase.           Decrease. 

Interest  on 
loan. 

Increase.     Decrease. 

Total  chai 
revenue  oi 
Net 
Increase. 

ge  to 
rate. 

Net 
Decrease. 

5  years 

133-73 

— 

—            

133-73 

— 

10  years 

133-73 

— 

70        — 

203-73 



8  years 

133-73 

— 

910        — 

1043-73 



6  years 

— 

55014 

—      1050 

— 

160014 

16  years 

107-85 

—        350 

457-85 

45  years 

The  principal  points  to  be  noticed  in  the  above  table  as 
compared  with  Table  XXXIIL  C,  which  shows  the  difference 
between  the  total  annual  loan  charges  under  the  original  con- 
ditions and  under  the  generally  adopted  equated  method,  are 
(1)  the  reduction  in  the  excess  additional  charge  during  the 
third  period  of  8  years  from  £2085-44  per  annum  to  £1043-73 
per  annum  due  solely  to  the  reduction  in  the  sinking  fund 
instalment,  and  (2)  the  additional  burden  imposed  upon  the 
first  15  years.  As  already  stated  it  has  been  assumed  that  it 
is  perfectly  equitable  to  consider  interest  upon  the  total  loan 
as  a  proper  charge  against  revenue  or  rate  during  the  later 
years,  although  the  increase  in  the  annual  interest  is  due  to  the 
delay  in  the  repayment  of  the  loan  for  purely  financial  reasons. 
The  following  statement  shows  the  final  repayment  of  the  loan 
by  means  of  the  instalments  in  Table  XXXIV.  C,  ascertained 
in  the  above  manner. 


THE    INCIDENCE    OF    TAXATION  427 


STATEMENT  XXXIV.  E. 

Loan  of  £56,000  (as  above). 

Showing  the  final  repayment  of  tlie  loan  by  the  operation  of 
the  sinking  fund  at  the  end  of  the  equated  period  of 
2o  years,  by  setting  aside  the  original  annual  instalments 
based  upon  the  life  of  the  asset,  and  a  further  additional 
instalment  spread  equally  over  the  equated  period.  Such 
instalments  are  shown  in  Table  XXXI\  .  C. 

1.  Amount  of  ii2o5098  per  annum  for  5  years     12482 
Amount  thereof  at  tiie  end  of  a  further 

18  years       —     21^49 

2.  Amount   of   £19T4'27    per   annum    for    10 

years    22633 

Amount  thereof  at  the   end   of   a  further 

8  years         28670 

3.  Amount    of    c£68o'87    per    annum    for    8 

years     6081 


Amount  of  loan     56000 


The  amended  annual  instalments  to  repay  £4340,  the 
balance  of  loan  unprovided  for  by  the  accumulation  of  the 
original  instalments  before  equation,  should  properly  be  dis- 
tributed over  the  equated  period  in  such  a  manner  that  they 
will  be  proportionate  to  the  original  instalments,  but  they  may 
be  spread  equally  over  the  equated  period  without  any  great 
injustice  being  caused.  Table  XXXIV.  C.  shows  that  the 
additional  annual  instalment  under  these  conditions  is  £133" 73. 
It  is,  however,  necessary  to  point  out  the  correct  method,  in 
order  that  it  may  be  applied  to  cases  where  the  magnitude  of  the 
loan  renders  it  desirable  to  make  an  absolute  equation  of  the 
incidence  of  the  sinking  fund  instalment,  as  well  as  of  the 
number  of  years  over  which  the  equated  burden  should  be 
■spread.  If  the  original  periodically  decreasing  annual  instal- 
ments are  set  aside  for  the  respective  repayment  periods,  and 
are  allowed  to  accumulate  until  the  end  of  the  equated  period 
they  will  provide  £51,660  of  the  original  loan  of  £56,000, 
leaving  £4,340  to  be  provided  by  the  accumulation  of  supple- 
mentary annual  instalments  to  be  set  aside  for  similar  numbers 
of  years  and  allowed  to  accumulate  for  the  same  periods.     The 


428    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

first  step  in  tlie  calculation  is  to  divide  the  £4,340  in  the  Bauie 
proportions  as  the  £51,660  as  follows  :  — 

(1)  5  years £20,Q40  £168:j-6 

(2)  10  years 26,728  2245-4 

(3)  8  years 4,892  411-0 


23  years £51,660  £4340 


The  component  parts  of  the  £4,340  represent  the  amounts 
which  will  be  in  the  fund  at  the  end  of  the  period  due  to  (1)  an 
annual  instalment  to  be  set  aside  for  5  years  and  then  accumu- 
lated for  a  further  18  years,  followed  by  (2)  an  annual  instal- 
ment to  be  set  aside  for  the  next  10  years  and  then  accumulated 
for  a  further  8  years,  followed  by  (3)  an  annual  instalment  to 
be  set  aside  for  the  final  8  years  at  the  end  of  which  period  the 
fund  will  mature.  This  latter  period  of  8  years  is  the  one 
which  bears  the  undue  burden  under  the  method  of  equation 
generally  adopted  and  which  it  is  the  object  of  the  present 
adjustment  to  remove.  Having  analysed  the  component  parts 
of  the  deficiency  of  £4,340,  the  respective  instalments  are 
ascertained  by  working  backwards.  In  the  case  of  item  (1)  it 
is  required  to  ascertain  the  amount  of  an  annuity  for  5  years, 
which  amount  if  accumulated  for  a  further  18  years  will 
provide  £1683-6.  The  first  step  therefore  is  to  find  by  standard 
calculation  form  No.  2  the  present  value  of  £1683-6  due  at  the 
end  of  18  years  at  3  per  cent.,  and  having  done  so  to  find  by 
standard  calculation  form  No.  3x  the  annuity  for  5  years  which 
will  amount  to  this  sum.  The  calculation  may  be  made  direct 
bv  Thoman's  method  as  follows:  — 


THE    INCIDENCE    OF    TAXATION  429 


CALCULATION  XXXIV.  G. 

To  fiud  the  annual  instalment  to  be  set  aside  and  accumulated 
for  a  given  number  of  years,  at  the  end  of  which  period 
the  amount  thereof  will  continue  to  accumulate  for  a 
further  specified  period  and  will  then  amount  to  a  given 
sum. 

Required,  the  annuity  for  5  years  which  will  amount  to  [the 
present  value  of  £1683-6  due  at  the  end  of  a  further  18 
years] . 

By  Thoman's  Tables  and  Logs. 

First  period  5  years.     Second  period  18  years. 

Log  of  the  given  future  sum     1683-60     3-2262330 

deduct  Log.  RN^  3  per  cent.  18  years  0-2310T00 


Loff  of       988-92     2-9951630 


-"to 


add  Log.  ft",  3  per  cent.  5  years      ...  93391623 


12-3343253 
deduct  Log.  RN^  3  per  cent.  5  years +  10  10-0641861 


Log  of  annuity  required      2-2701392 

Annual  instalment  required     . . .  £186-26 


Note.     This  calculation  may  be  compared  with  XYI.  D.  1 
and  XXYII.  C. 


The  second  item  may  be  ascertained  in  a  similar  manner, 
but  the  third  calculation  consists  merely  of  finding  the  annual 
instalment,  and  it  may  be  performed  on  standard  form  No.  3x. 
It  is  not  necessary  to  give  the  actual  details  of  the  calculations, 
but  merely  to  state  the  results  in  the  following  table  which 
shows  the  manner  in  which  the  above  deficiency  of  £4,340  will 
be  provided  at  the  end  of  the  equated  period. 


430  REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 


STATEMENT  XXXIV.  H. 

Loan  of  £56,000  (as  above). 

Showing  the  supplementary  annual  instalments  to  be  set  aside 
and  added  to  the  sinking  fund  during  the  equated  period 
of  23  years,  to  repay  £4,340  of  the  original  loan  un- 
provided by  the  original  annual  instalments  added  to  the 
fund. 

The  following  supplementary  annual  instalments  are 
proportionate  to  the  original  annual  instalments  based 
upon  the  life  of  the  asset,  and  are  not  equal  during  the 
whole  of  the  repayment  period,  as  was  the  case  in  Table 
XXXIV.  C,  and  Statement  XXXIV.  F. 

1.  Amount  of   £186-26   per    annum    for   5 

years     ..." 988'92 

Amount  thereof  at  the  end  of  a  furtber 

18  years.        Calculation  XXXIV.  G.     1683-60 

2.  Amount  of  £154-62  per  annum   for   10 

years 177260 

Amount  thereof  at  the  end  of  a  further 

8  years         2245-40 

3.  Amount  of  £4622  per  annum  for  8  years  41100 

4340-00 


The  above  annual  instalments  may  be  usefully  compared 
with  those  previously  obtained  where  the  supplementary  annual 
instalment  of  £133-73  is  spread  equally  over  the  equated  period, 
as  shown  in  Table  XXXIV.  C.  The  following  table  shows 
the  animal   iiistnlmeiits  under  the  present  method:  — 


THE    INCIDENCE    OF    TAXATION  431 


TABLE  XXXIV.  J. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature 
having  prescribed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Showing  the  annual  charges  to  revenue  or  rate  in  respect  of 
the  annual  instalments  under  (1)  the  equated  method 
generally  adopted,  and  (2)  in  which  the  annual  instalments 
as  originally  ascertained  are  supplemented  by  additional 
annual  instalments  spread  over  the  equated  j^eriod  in  pro- 
portion to  the  original  periods  allowed. 


riods  of  equal 
incidence. 

Original 

annual 

instalment. 

Additional 

annual 
instalment. 

Total 

annual 

instalment. 

Annual 
instalment 
as  equated. 

5  years 

2217-25 

186-26 

2403-51 

1725-58 

10  years 

1840-54 

154-62 

1995-16 

1725-58 

8  years 

550-14 

46-22 

596-36 

1725-58 

23  years 


It  is  not  necessary  to  give  details  of  the  actual  calculations, 
but  the  following  statement  has  been  prepared  in  order  to  show 
the  final  repayment  of  the  loan  at  the  end  of  the  equated  period 
of  23  years,  by  means  of  the  annual  instalments  in  Table 
XXXIV.  J.,  thereby  proving  the  accuracy  of  the  above  method. 


432  REPAYMENT    OF    LOCAL   AND    OTHER   LOANvS 


STATEMENT  XXXIY.  K. 

Loan  of  £56,000  (as  above). 

Showing"  the  final  repayment  of  the  loan  by  the  operation  of 
the  sinking  fnnd  at  the  end  of  the  equated  period  of 
23  years  by  annual  instalments  spread  over  the  equated 
period,  with  due  regard  to  the  life  of  the  asset  instead  of 
being  spread  equally  over  such  period. 

Table  XXXIY.  J. 

1.  Amount   of  <£2403'51   per  annum   for  5 

years      127606 

Amount  thereof  at  the  end  of  a  further 

18  years         — 21724 

2.  Amount  of  ±*1995'16  per  annum  for  10 

years      22872'3 

Amount  thereof  at  the  end  of  a  further 

8  years 28973 


3.  Amount   of   £59636   per   annum   for   8 
years      


5303 


Amount  of  loan 56000 


Four  methods  have  now  been  shown  by  which  the  loan  of 
£56,000  may  be  repaid  before  and  after  the  eqiuition  of  the 
period,  and  ihe  total  annual  loan  charges  for  instalment  and 
interest,  under  each  method,  will  uow  be  summarised. 


THE    INCIDENCE    OF    TAXATION 


433 


TABLE  XXXIV.  L. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature 
having  prescribed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Comparison  of  the  total  annual  charges  to  revenue  or  rate 
during  the  several  periods  of  equal  incidence  forming  part 
of  the  original  extended  period  of  repayment,  and  now 
constituting  the  equated  period,  under  the  following 
methods  :  — 

1.  The  generally  adopted  method  of  equation  in  which 

the  instalment  is  spread  equally  over  the  equated 
period. 

2.  The  instalments  originally  calculated  before  equation 

based  upon  the  life  of  the  asset. 

3.  The  method  in  which  the  original  annual  instalments, 

based  upon  the  life  of  the  asset,  are  set  aside  during 
the  equated  period,  and  any  deficiency  is  made  good 
by  a  supplementary  annual  instalment  spread 
equally  over  the  equated  period. 

Table  XXXIY.  C. 

4.  The   method    in   which   the    instalments    during   the 

equated  period  are  exactly  proportional  to  the  life 
of  the  individual  assets  and  to  the  original  annual 
instalments  based  thereon.  Table  XXXIY.  J. 


Annual  instalments. 


Period  of  equal 
incidence. 

5  years 

10  years 

(S  vears 


Tlie  equated 
method. 

1725-58 

1725-58 
1725-58 


Method 

(2) 
above 

Table 
XXXIII,  A. 

2217-25 

1840-54 
550-14 


Method 

(3) 
above 

Table 
XXXIV.  c. 

2350-98 
1974-27 

683-87 


Method 

(4) 
above 

Table 
XXXIV.  J. 

2403-51 

199516 

596-36 


Interest  upon  the  loan. 


See  Table  XXXIII.  B. 


Periods  of 
equal  incidence 

The  equated 
method 

Method 

(2) 
above. 

Method 

(3) 
above. 

Method 

(4) 
above. 

5  years 

1960 

1960 

1960 

1960 

10  years 

1960 

1890 

1960 

1960 

8  years 

1960 

1050 

1960 

1960 

AID 


434 


REPAYMENT    OF   LOCAL   AND    OTHER   LOANS 


Total  annual  charges  for  instalment  and  interest  on  loan. 


Periods  of 

equal 
incidence. 

5  years 

Instalment 
Interest 

Total 

Instalment 
Interest 

Total 

Instalment 
Interest 

Total 

Instalment 
Interest 

Total 

Instalment 
Interest 

Total 

1. 

The  equated 
method. 

1725-58 
1960-00 

Method 

(2) 
above. 

2217-25 
196000 

Method 

(3) 
above. 

2350-98 
196000 

Method 

(4) 
above. 

2403-51 
196000 

3685-58 

4177-25 

4310-98 

4363-51 

0  years 

1725-58 
1960-00 

1840-54 
189000 

1974-27 
1960-00 

1995-16 
1960-00 

3685-58 

3730-54 

3934-27 

3955-16 

8  years 

1725-58 
196000 

550-14 
1050-00 

683-87 
1960-00 

596-36 
1960-00 

3685-58 

1600-14 

2643-87 

2556-36 

6  years 

— 

550  14 
105000 

— 

— 

— 

1600-14 

— 

— 

.6  years 

— 

107-85 
350-00 

— 

— 

— 

457-85 

— 

— 

The  conclusions  to  be  drawn  from  the  above  results  are,  that 
the  generally  adopted  method  of  equating  the  burden  upon 
successive  generations  of  ratepayers  is  unjust  to  the  later  years 
of  the  equated  period  seeing  that  the  acceleration  of  the  final 
repayment  of  the  loan  ought  to  impose  a  larger  burden  upon 
each  year  of  the  equated  period.  It  is  also  obvious  that  on  the 
contrary  the  generally  adopted  method  of  equation  relieves  the 
earlier  years  instead  of  increasing  the  annual  charge  during 
such  years.  The  dates  of  repayment,  as  originally  fixed  before 
equation,  were  based  upon  the  life  of  the  asset,  and  this  should 
not  be  lost  sight  of  in  amending  the  annual  instalment  after 
the  equation  of  the  period,  as  it  is  in  fact  ignored,  in  the 
generally  adopted  method  in  which  the  whole  of  the  outlay  is 
treated  as  having  an  equal  repayment  period.  The  proper  and 
consistent  method  of  apportioning  the  burden  between  the 
several  years  of  the  <M]uated  period  is  to  adhere  as  closely  as 


I'll 


THE    INCIDENCE    OF    TAXATION  435 

possible  to  the  original  annual  instalments  before  equation 
because  by  this  means  alone  can  two  important  results  be  attained, 
namely,  the  annual  redemption  charge  to  the  revenue  or  rate 
account  of  each  year  will  be  proportionate  to  the  wastage  of  the 
asset  and,  which  is  equally  important,  the  loan  in  respect  of 
outlay  of  short  duration,  if  not  actually  repaid,  will  be  in  the 
sinking  fund  slightly  earlier  than  the  date  at  which  it  would 
have  been  repaid  under  the  original  conditions.  This  will 
remove  an  objection  at  present  existing  as  to  re-borrowing  for 
outlay  of  short  duration  where  the  repayment  of  the  loan  has 
been  delayed  by  the  equation  of  the  period.  A  final  conclusion 
seems  to  be  that  in  making  the  adjustment  in  the  annual  instal- 
ment the  true  mathematical  method  last  described  should  be 
followed  although  it  may  involve  rather  more  intricate  calcula- 
tions. The  equation  of  the  incidence  of  the  annual  interest 
charges  will  be  considered  in  the  following  chapter. 


436         REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 


CHAPTER  XXXV. 

THE  EQUATION  OF  THE  INCIDENCE  OF  TAXATION 

[continued) . 

INTEREST  UPON  THE  LOAN. 

The  method  of  adjusting  the  annual  charges  to  revenue 
OR  rate  during  the  equated  period  in  proportion  to 

THE  LIFE  OR  DURATION  OF   CONTINUING  UTILITY  OF  THE  ASSET 
CREATED  OUT  OF  THE  LOAN. 

By  CHARGING  THE  REVENUE  OR  RATE  ACCOUNT  OF  EACH 
YEAR  OF  THE  EQUATED  PERIOD  WITH  THE  ANNUAL  AMOUNT  OF 
INTEREST  PAYABLE  BEFORE  EQUATION,  AND  IN  ADDITION 
THERETO  A  SUPPLEMENTARY  ANNUAL  AMOUNT  PROPORTIONATE 
YEAR  BY  YEAR  TO  THE  ANTSTUAL  INTEREST  CHARGES  BEFORE 
EQUATION. 

A    GENERAL    SUMMARY    OF    THE    RESULTS    OBTAINED    IN    CHAPTERS 

XXXIII,  XXXIY  AND  XXXY. 


The  previous  enquiry  into  tlie  method  of  adjusting  the 
annual  incidence  of  the  h)an  burden,  after  equation,  is  confined 
solely  to  the  annual  instalment  to  be  charged  to  revenue  or  rate 
and  added  to  the  sinking  fund.  The  result  of  the  enquiry  is  to 
prove  that  the  final  8  years  of  the  equated  period  are  made  to 
bear  not  only  the  whole  of  the  relief  to  the  22  years  of  the 
original  period  beyond  the  equated  period,  but  a  certain  amount 
in  relief  of  the  earlier  years  of  the  equated  period.  The  annual 
instalments  during  the  final  8  years  of  the  equated  period  are 
as  follows  :  — • 

Before  equation,  under  the  original  conditions 

Table  XXXIII.  A.       £55014 


After  equation  : 

(1)  by  the  method  generally  adopted 

Table  XXXIII.  A.     £172558 


(2)  by  tlie  true  method  just  described 

Table  XXXIV.  J.       £596-36 


THE    INCIDENCE    OF    TAXATION  437 

The  annual  charge  ior  interest  upon  the  loan  during  the 
final  8  years  of  the  equated  period  has  already  been  referred  to. 
Under  the  original  conditions,  before  equation,  this  annual 
charge  was  £1,050  only,  owing  to  the  fact  that  £26,000  of 
loan  had  been  repaid  by  the  end  of  the  15th  year.  Under  the 
equated  method  the  whole  of  the  loan  is  not  repayable  until  the 
end  of  the  equated  period,  and  consequently  the  interest  upon 
the  total  loan  becomes  an  annual  charge  against  revenue  or  rate 
during  the  whole  of  the  equated  period.  The  effect  is  to 
impose  upon  the  final  8  years  of  the  equated  period  an  interest 
burden  of  £910  per  annum  in  addition  to  the  annual  charge  of 
£1,050  under  the  original  conditions.  The  first  period  of 
5  years  is  charged  annually  with  the  same  amount  of  interest 
upon  the  loan  under  both  methods;  and  the  second  period  of 
10  years  is  charged  with  an  additional  £70  per  annum  only, 
being  interest  upon  £2,000  of  loan  which  would  otherwise  have 
been  repaid  at  the  end  of  the  first  period  of  5  years.  It  is 
therefore  apparent  that  the  final  8  years  of  the  equated  period 
bears  the  greater  portion  of  the  interest  of  which  the  final 
22  years  of  the  original  period  is  relieved,  in  addition  to  the 
whole  of  the  sinking  fund  instalment. 

The  interest  charges  against  the  revenue  or  rate  accounts 
of  the  final  8  years  of  the  equated  period  are  as  follows  :  — 

Before  equation,  under  the  original  conditions 

Table  XXXIII.  B.     £1050 

After  equation,  under  the  method  generally 
adopted,  and  also  under  the  true  method, 
above  described,  if  limited  to  the  annual 
instalment  only  Table  XXXIY.  L.     £1960 


The  total  annual  loan  charges  during  the  final  8  years  of 
the  equated  period  are  therefore  as  follows  (Table  XXXIV.  L.)  : 

Before  equation,  under  the  original  conditions  : 

Instalment         £550-14 

Interest      ^lO^OOO 

£160014 


43S    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

After  equation :  — 

(1)  by  the  method  generally  adopted  : 

Instalment         £172558 

Interest      £196000 


method     just 
the     annual 

described, 
instalment 

.     £596-36 
.  £196000 

(2)  by     the     true 
relating     to 
only: 
Instalment 
Interest 

£?5iifi-3fi 

The  method  just  described  removes  the  injustice  to  the  final 
3'ears  of  the  equated  period  so  far  as  the  annual  instalment  is 
concerned,  but  leaves  untouched  the  question  of  the  interest 
upon  the  loan. 

The  advisability  of  making  a  similar  adjustment  with  regard 
to  the  interest  charges  is  a  matter  upon  Avhich  opinion  may  be 
divided,  and  in  order  to  elucidate  the  question  it  is  necessary  to 
state  a  few  general  propositions.  The  practice  now  adopted  in 
the  case  of  all  original  loans  is  to  spread  the  repayment 
of  the  princijjal  over  a  series  of  years  commensurate  to  the  life 
or  period  of  continuing  utility  of  the  asset  created  out  of  the 
loan,  with  certain  limitations  as  to  assets  of  long  continuing  or 
permanent  utility.  Of  the  three  alternative  methods  allowed 
by  statute,  only  one,  the  instalment  method,  involves  an  un- 
equal annual  charge  in  respect  of  interest  upon  the  loan.  In 
the  case  of  the  annuity  and  sinking  fund  methods  the  total 
annual  charges  for  principal  and  interest  are  equal  throughout 
the  period  of  repayment  or  redemption.  But  in  both  these 
cases  interest  is  payable  during  such  redemption  period  only, 
and  of  course  ceases  on  the  final  repayment  of  the  loan.  In 
other  Avords,  the  total  annual  loan  charges,  both  for  instalment 
and  interest  are  spread  over  a  period  depending  upon  the  life 
of  tlie  asset,  and  the  annual  charge  to  revenue  or  rate  is  the 
same  in  each  year  of  the  period.  This  principle  is  carried  out, 
and  is  considered  equitable,  in  the  case  of  individual  loans 
relating  to  outlay  of  one  character  only,  repayable  on  fixed 
dates,  and  in  respect  of  which  separate  sinking  funds  are  or 
may  be  kept. 

But  the  conditions  are  different  in  the  case  of  the  equation 
of  the  period  of  repayment  whether  the  equation  is  made  on  the 


THE    INCIDENCE    OF    TAXATION  439 

consolidation  of  existing  loans  having  varying  unexpired 
periods  of  repayment,  or  whether  it  is  made  in  respect  of  one 
loan,  relating  to  outlay  of  a  varied  character,  each  class  having 
different  periods  of  continuing  utility  and  consequent  periods 
of  repayment.  The  effect  of  the  equation  of  the  period  as 
regards  the  loan  holder  has  already  been  fully  discussed  in 
Chapter  XXXII,  where  it  has  been  ascertained  that  the 
arithmetical  method  of  finding  the  equated  period  generally 
adopted,  although  wrong  in  principle,  may  perhaps  be  con- 
sidered sufficiently  correct. 

Prior  to  an  equation  due  to  either  of  the  events  already 
described,  the  amount  to  be  charged  to  the  rate  or  revenue 
account  of  each  year  has  already  been,  or  may  be,  ascertained, 
and  the  result  of  any  true  equation  should  be  that  the  future 
substituted  total  annual  burdens  are  in  proportion  to  the 
original  obligations;  any  variations  therefrom  being  due  only 
to  the  substituted  period  imposed,  and  since  the  original  annual 
burdens  were  not  equal  during  the  original  periods  they 
certainly  should  not  be  equal  during  the  substituted  period. 

The  equation  of  the  incidence  of  the  sinking  fund  instalment 
has  been  fully  discussed  in  Chapter  XXXIV,  and  a  method 
described  in  detail  of  finding  the  future  instalments  only.  If 
it  be  required  to  adjust  the  annual  incidence  of  the  interest 
upon  the  loan  with  a  similar  object  in  view,  the  adjustment 
cannot  be  made  in  quite  the  same  manner,  seeing  that  in  this 
case  an  equal  annual  amount  of  interest  has  to  be  paid  to 
the  loan  holder,  but  the  burden  of  providing  the  interest  has  to 
be  spread  unequally  over  the  period.  The  method  is  in  fact  a 
combination  of  the  sinking  fund  and  annuity  methods  of 
repayment  of  debt,  with  varying  instead  of  equal  annual 
charges  to  revenue  or  rate. 

In  order  to  simplify  the  method  of  adjustment  and  to  bring 
it  as  near  as  possible  to  the  method  of  equating  the  annual 
incidence  of  the  sinking  fund  instalment,  already  described  in 
Chapter  XXXIV,  it  is  necessary  to  reduce  the  interest  upon  the 
loan  under  both  sets  of  conditions  to  its  corresponding  capital 
value  at  the  end  of  the  equated  period  of  23  years.  The  capital 
value  might  also  be  expressed  in  terms  of  the  present  value,  but 
it  would  not  be  so  convenient  because  the  loan  is  repayable  at  a 
future  date.  The  annual  interest  of  £1,960,  payable  upon  the 
total  loan  during  the  23  years  of  the  equated  period,  will  there- 
fore be  treated  as  if  it  were  allowed  to  accumulate  at  compound 
interest  at  3  per  cent,  until  the  end  of  that  period.  The  sum 
to  which  £1,960  per  annum  will  then  amount  may  be  ascer- 


440    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

tained  in  the  usual  way  by  standard  calculation  form,  No.  3, 
and  will  be  found  to  be  £6360770. 

For  the  present  purpose  this  amount  may  be  treated  in 
exactly  the  same  manner  as  the  total  loan  of  £56,U0U  when 
adjusting  the  annual  incidence  of  the  sinking  fund  instalment 
only.  In  the  same  way  that  it  was  there  required  to  find  three 
annual  instalments  of  unequal  amount,  to  be  set  aside  and 
accumulated  for  successive  periods  of  5,  10,  and  8  years 
resjjectively,  it  is  now  required  to  find  three  annual  amounts  of 
interest  to  be  charged  to  the  revenue  or  rate  account,  such 
annual  interest  charges  to  be  of  vmequal  amount  during  each 
of  the  above  periods,  and  to  bear  a  relation  to  the  life  of  the 
asset  as  expressed  in  the  repayment  periods  originally  pre- 
scribed. 

In  the  case  of  the  adjustment  of  the  annual  instalment  the 
basis  of  the  calculation  was  the  original  annual  instalments 
before  equation  and  in  the  present  example  the  basis  is  the 
original  charges  for  interest  upon  the  loan.  In  this  connection 
it  is  important  to  point  out,  as  will  be  seen  on  reference  to  the 
second  column  in  the  third  part  of  Table  XXXIV.  L.,  that 
there  is  not,  before  equation,  any  definite  ratio  between  the 
annual  instalment  and  the  annual  interest  charges  during  any 
period  of  equal  incidence,  consequently  the  two  adjustments  are 
quite  distinct. 

The  next  step  is  to  ascertain  the  accumulated  amount  at  the 
end  of  23  years  of  the  original  annual  interest  charges  before 
equation  shown  in  Table  XXXIII.  B.  in  the  same  way  that  the 
amount  in  the  sinking  fund  was  ascertained  at  the  same  date 
by  the  accumulation  of  the  original  annual  iuvstalments  shown 
in   Statement   XXXIV.  B.   as  follows:  — 


TABLE  XXXV.  A. 

Loan  of  £56,000  (authorised  for  outlays  of  varying  nature 
liaving  prescribed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Showing  the  accumulated  iimount  at  the  end  of  the  e(in;\1ed 
period  of  23  years,  of  the  original  annual  interest  charges, 
as  shown  in  Table  XXXIII.  B.     (Interest  at  3  per  cent.) 

This  statement  should  be  compared  with  XXXIV.  B. 


THE    INCIDENCE    OF    TAXATION  441 

(Ij  Amount  of  i>l,9GU  per  annum  for  5 

years       '...     10405-90 

Amount  thereof  at  tlie  end  of  a  further 

18  years         17715-40 

(2)  Amount  of  £1,890  per  annum  for  10 

years       2166670 

Amount  thereof  at  the  end  of  a  further  , 

8  years 27446-80 

(3)  Amount  of  £1,050  per  annum   for   8 

years       9336-90 


Total,  being'  the  accumulated  amount  of  the  original 
annual  interest  charges,  before  equation,  at  the 
end  of  the  equated  period 54499-10 


Up  to  this  point  it  has  been  ascertained  that  the 
accumulated  amount,  at  the  end  of  the  equated 
period  of  23  years,  of  the  equal  interest 
charges  of  £1,960  per  annum  after  equation  is     £6360770 

and  the  corresponding  amount  of  the  original 
varying  annual  interest  charges  as  shown  by 
the  foregoing  table     £5449910 


A  deficiency  of  ... 


£9108-60 


which  is  exactly  comparable  with  the  deficiency  of  £4,340  m 
the  case  of  the  annual  instalment  (after  Statement  XXXIV.  B.). 
The  adjustment  of  the  present  deficiency  may  be  made  by 
the  method  described  in  Chapter  XXXIV  leading  up  to 
Calculation  XXXIV.  G.,  and  Statement  XXXIV.  H.,  but  as 
the  conditions  as  to  period  and  rate  per  cent,  are  similar  in 
both  cases,  and  differ  only  in  amount  it  is  possible  to  adopt  a 
shorter  method  by  utilising  the  information  there  obtained 
and  increase  the  supplementary  annual  charges  found  in 
Statement  XXXIV.  H.,  in  the  ratio  that  4340  bears  to  9108-60 
as  follows  :  — 


442 


REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


TABLE  XXXV.  B. 

Showing  tlie  method  of  finding  the  .supplementary  annual 
charges  to  revenue  or  rate  to  be  added  to  the  original 
annual  interest  charges  before  equation. 


Periods  of  equal  incidence. 

Amount  of  deficiency... 


Deficiency  at  end  of  23  years. 
Annual  instalment.         Interest  on  loan. 
Statement  XXXIV.  H.  As  above. 


Supplementary  annual  charges  to 
revenue  or  rate. 


5  year  period 

10  year  period 

8  year  period 


23  years 


Total 


434000 


9108-60 


Annual  Annual 

instalments  interest  charges 

Table  XXXIV.  J.     Table  XXXV.  C. 


186-26  390-93 

154-62  324-51 

46-22  9700 


387-10 


812-44 


The  above  annual  interest  charges  are  ascertained  from 
the  amounts  in  the  first  column  by  the  ordinary  rules  of  pro- 
portion or  by  logs.,  of  which  it  is  not  necessary  to  show  the 
actual  working.  The  total  annual  interest  charges  to  revenue 
or  rate  during  the  equated  period  may  now  be  stated  in  the 
following  table  :  — 


THE    INCIDENCE    OF    TAXATION 


443 


TABLE  XXXV.  0. 

Loan  of  i;56,000  (authorised  for  outlays  of  varying  nature 
having  prescri])ed  periods  of  repayment),  the  whole  to  be 
redeemed  in  one  sum  at  the  end  of  an  equated  period. 

Showing  the  annual  charges  to  revenue  or  rate  in  respect  of 
interest  upon  the  loan  under  (1)  the  equated  method 
generally  adopted,  and  (2)  in  which  the  annual  interest 
charges  originally  payable  are  supplemented  by  additional 
annual  amounts  spread  over  the  equated  period  in  propor- 
tion to  the  original  interest  obligations. 

This  table  should  be  compared  with  Table  XXXIV.  J. 


Equated  annual  interest  charges. 


Periods 
of 
ecjual  incidence. 

Original 

annual 

interest  charges. 

Additional 

annual 

interest  charges. 

Total 

annual 

interest  charges. 

Annual 

interest  charges 

under  the 
equated   method. 

5  years 

1960-00 

390-93 

2350-93 

196000 

10  years 

189000 

324-51 

2214-51 

196000 

8  years 

105000 

97-00 

1147-00 

196000 

23  years 

Statement  XXXIV.  K.  shows  the  final  repayment  of  the 
loan  by  means  of  the  amended  annual  instalments  to  be  spread 
over  the  equated  period  with  due  regard  to  the  life  of  the  asset, 
instead  of  being  spread  equally  over  such  period,  and  thereby 
proves  the  accuracy  of  the  method  adopted  with  regard  to  the 
annual  instalment.  In  a  similar  manner,  although  expressed 
in  different  terms,  the  following  Statement  XXXY.  D.  proves 
the  accuracy  of  the  method  adopted  in  order  to  equate  the 
incidence  of  the  annual  interest  charges. 


444         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 

STATEMENT  XXXV.  D. 

Loan  of  £56,000  (as  above). 
Sliowiiig-  that  the  accumulated  amount  oi  the  amended  annual 
interest  ckarges  ascertained  as  in  Table  XXXV.  C.  will  be 
equal  to  the  accumulated  amount  of  the  equal  annual 
interest  charges  after  equation,  both  at  the  end  of  23  years, 
at  3  per  cent,  per  annum. 

This  statement  should  be  compared  with  Statement  XXXIV.  K. 

(1)  Amount  of  £2c550'9o   jjer   annum   for 

5  years  

Amount  thereof  at  the  end  of  a  further 

18  years         21248-80 

(2)  Amount  of  £2214"51   per   annum   for 

10  years  

Amount  thereof  at  the  end  of  a  further 

8  years 32159-40 

(3)  Amount   of   £1147   per   annum   for   8 

years       1019950 


which  is  the  accumulated  amount  of  an  annuity  of 
£1960  for  23  years  at  3  per  cent,  as  previously 
ascertained  £63607-70 


The  above  calculations  (1)  and  (2)  have  been  made  direct 
by  the  "  method  by  step  "  shown  in  Statement  XXVII.  C. 

It  will  be  gathered  from  the  above  Statement  XXXV.  D., 
that  tlie  annual  interest  charges  to  revenue  or  rate  during  the 
first  two  periods  of  5  years  and  10  years  are  greater  than  the 
annual  amounts  of  interest  payable  to  the  loanholders  during 
those  periods  after  equation,  as  follows  :  — - 

5  years  (2350-93—1960)   an  increase  of  390-93 
10  years  (2214-51—1960)  an  increase  of  254-51 

and  that  the  annual  amounts  of  interest  payable  to  the  loan- 
holders  during  the  final  8  years  of  the  equated  period  are 
greater  than  the  amended  annual  amounts  charged  to  revenue 
or  rate  during  that  period,  in  Statement  XXXV.  D.,  as  follows  : 

8  years  (1960—1147),  a  decrease  of  81300. 


THE    INCIDENCE    OF    TAXATION  445 

The  correctness  of  the  foregoing  calculations  is  proved  by  the 
following  statement  giving  the  accumulated  amounts  of  the 
above  annuities  at  the  end  of  the  2ord  year,  without  details  of 
the  actual  calculations  which  are  similar  to  XXVII.  C. :  — 

Amount  of  £390-93  per  annum  for  5  years,  accu- 
mulated for  a  further  period  of  18  years  at  3 
per  cent,  per  annum     £3533'40 

Amount  of  £254-51  per  annum  for  10  years,  accu- 
mulated for  a  further  period  of  8  years,  at  3 
per  cent,  per  annum     


£369600 


£7229-40 
which  is  equal  to  the 

Amount  of  £813*00  per  annum  for  8  years  at  3  per 

cent,   per  annum £(-^^9  41) 

This  proves  that  the  amounts  charged  to  the  revenue  or  rate 
account  during  the  first  15  years,  in  excess  of  the  amounts 
annually  payable  to  the  loanholders  during  that  period,  will, 
if  accumulated,  be  suflacient  to  provide  the  future  annual 
deficiencies  in  the  amounts  charged  to  revenue  or  rate  account 
during  the  final  8  years  of  the  equated  period.  It  also  points 
out  the  methods  to  be  adopted  as  regards  the  actual  book- 
keeping, and  indicates  the  opening  of  an  account  which  may 
be  termed  an  :  — 

"  Equated  Loan  Interest,   Reserve  Account," 

and  which  will  closely  resemble  the  repayment  of  a  loan  by  an 
equal  annual  instalment  of  principal  and  interest  combined, 
or  the  annuity  method,  but  with  a  varying  instead  of  an  equal 
annual  charge  to  revenue  or  rate.  To  the  extent  that  the 
annual  charges  to  revenue  or  rate  are,  during  the  earlier  years, 
greater  than  the  annual  amounts  payable  to  the  loanholders  by 
way  of  interest,  the  account  will  also  partake  of  the  nature  of 
a  sinking  fund,  and  will  therefore  require  the  same  careful 
future  supervision  as  to  the  amount  standing  to  the  credit  of 
the  account,  the  rate  of  accumulation,  and  also  the  immediate 
preparation  of  a  pro  forma  account  showing  the  ultimate  work- 
ing out  of  the  account. 

As  regards  the  actual  book-keeping  the  above  interest  reserve 
account    may   be   treated    in   two    distinct   ways,    namely,    by 


446  REPAYMENT   OF   LOCAL   AND    OTHER    LOANS 

crediting  the  account  with  the  total  annual  amounts  of  interest 
charged  to  revenue  or  rate,  as  shown  in  Table  XXXY.  C.  and 
debiting  the  account  with  the  annual  interest  payable  to  the 
loanholders,  which  is  the  more  scientific  method  as  yielding  an 
exact  record  of  the  actual  transactions.  The  other  method  is 
to  treat  it  as  a  reserve  account  pure  and  simple  and  credit  it 
only  with  the  above  excess  annual  amounts  of  £39093  and 
<£254"21  charged  to  revenue  or  rate  account  during  the  periods 
of  5  and  10  years  respectively.  During  the  third  period  of 
8  years  the  interest  reserve  account  would  of  course  be  debited, 
and  the  revenue  or  rate  account  credited,  with  the  difference 
of  £813  per  annum  already  referred  to.  Unlike  a  sinking  fund 
proper,  the  amount  to  the  credit  of  the  interest  reserve  account 
need  not  be  separately  invested,  but  may  be  merged  in  the 
general  assets  of  the  undertaking,  provided  always  that  the 
proper  annual  amounts  of  interest  at  the  calculated  rate  of 
accumulation  are  credited  to  the  account,  and  charged  to  the 
current  year's  revenue  or  rate  account.  If  the  account  be  kept 
in  this  manner  and  compared  annually  with  the  pro  forma 
account  there  should  not  arise  at  any  time  any  necessity  to 
make  an  adjustment  so  long  as  the  repayment  period  remains 
unaltered.  The  following  pro  forma  account  will  illustrate  the 
method  of  keeping  the  interest  reserve  account  applicable  to  the 
foregoing  example  :  — 


THE    INCIDENCE    OF    TAXATION  447 

TABLE  XXXV.  E. 

Equated  Loan  Interest,     Reserve  Account. 

Interest,  3  per  cent,  per  annum. 


Amount  to 

Interest 

Interest 

credit  at 

charged  to 

paid  to 

Balance 

beginning 

Interest 

revenue 

Total 

loan 

carried 

of  year. 

thereon. 

or  rate. 

credits. 

holders. 

forward. 

Year. 

1  Nil          Nil  2350-93  2350-93  196000  39093  1 

2  390-93       11-73  2360-93  2753-59  196000      79359  2 

3  793-59      23-81  235093  316833  196000  1208-33  3 

4  1208-33  36-25  235093  3595-51  196000  163551  4 

5  1635-51  49-06  235093  403550  196000  207550  5 

6  2075-50  62-26  2214-51  435227"  1960-00  2392-27  6 

7  2392-27       7177  221451  467855  196000  271855  7 

8  2718-55  81-56  221451  5014-62  1960-00  3054-62  8 

9  3054-62  91-64  2214-51  536077  196000  3400-77  9 

10  3400-77  102-02  221451  571730  1960-00  375730  10 

11  3757-30  112-72  221451  608453  196000  412453  11 

12  4124-53  123-73  221451  646277  196000  450277  12 

13  4502-77  135-08  221451  685236  196000  489236  13 

14  4892-36  14677  2214-51  7253-64  1960-00  5293-64  14 

15  5293-64  15885  2214-51  766700  1960-00  5707-00  15 

16  570700  171-21  114700  702521  196000  506521  16 

17  5065-21  151-96  114700  636417  1960-00  4404-17  17 

18  4404-17  132-12  114700  568329  196000  372329  18 

19  3723-29  11170  114700  498199  1960-00  3021-99  19 

20  3021-99  90-66  114700  425965  1960-00  229965  20 

21  2299-65  6899  114700  351564  1960-00  1555-64  21 

22  1555-64  4667  114700  274931  196000  789-31  22 

23  789-31  23-69  1147-00  196000  1960-00        Nil  23 


448    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

A  comparison  lias  already  been  made,  in  Table  XXXIV.  L., 

between    the    total    annual    loan    charges    under    the    original 

conditions    and    under    the    generally    adopted    method    after 

equation;   and  a  final  comparison  will  now  be  made  between 

those    methods    and    the    one    just    described.     The    following 

results  are  worthy  of  careful  study  and  show  the  very  wide 

difference    between    the    incidence    of    the    total    annual    loan 

burden  under  the  generally  adopted  method  after  equation  on 

the  one  hand  as  compared  with  the  annual  charges  under  the 

original  conditions  before  equation  and  also  under  the  amended 

method  just  described.     Under  the  method  generally  adopted 

the  annual  burden,  both  as  regards  the  annual  instalment  and 

interest  upon  the  loan,  is  spread  equally  over  the  whole  of  the 

equated  period  with  a  total  disregard  to  the  life  of  the  asset  and 

the    consequent    repayment   periods    originally   based   thereon. 

The  following  table  shows  that  the  original  incidence  of  the 

burden  is  departed  from  under  the  generally  adopted  method 

after  equation  in  that  it  relieves  the  earlier  years,  and  throws 

a  severe  additional  burden  upon  the  later  years,  of  the  equated 

period.     The  result  of  the  author's  method  is  that  the  revenue 

or  rate  account  of  each  year  of  the  equated  period  is  charged 

with    an    amount    in   respect   both   of    annual   instalment   and 

interest  upon  the  loan,  which  is  exactly  in  proportion  to  the 

amount   with   which  it  would   have   been   charged   under   the 

original  conditions.    These  amended  annual  charges  are  greater 

than  under  the  original  conditions  and  include,  as  is  equitable, 

the  relief  to  the  post  equated  period,  and  such  relief  is  imposed 

rateably  upon  each  year  of  the  equated  period  instead  of  being 

charged  against  the  later  period  of  8  years  only. 


THE    INCIDENCE    OF    TAXATION 


449 


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450    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

In  order  to  make  tlie  foregoing  results  perfectly  clear,  three 
charts  have  been  prepared  as  follows  :  — 

The  Equation  of  the  Incidence  of  Taxation.  Chart.  I. 

Showing  the  total  annual  loan  charges  in  respect  of  the  sinking 
fund  instalment  and  interest  upon  the  loan,  during  each 
year  of  the  original  and  equated  periods. 

(A).  Under  the  original  conditions,  before  equation. 
(B).   Under  the  equated  method,  as  generally  adopted. 

(C).  Under  the  author's  method  of  equation  relating  to 

the    annual    instalment    only,     as    described    in 

Chapter  XXXIV. 
(D).  Under  the  author's  method  of  equation  relating  to 

the  annual  instalment  and  interest  upon  the  loan, 

as  described  in  Chapter  XXXV. 

The  Equation  of  the  Incidence  of  Taxation.  Chart.  II. 

Showing  the  total  annual  loan  charges  in  respect  of  the  sinking 
fund  instalment  and  interest  upon  the  loan,  during  each 
year  of  the  original  and  equated  periods. 

(A).  Under  the  original  conditions,  before  equation. 
(B).  Under  the  equated  method,  as  generally  adopted. 
(D).  Under  the  author's  method  of  equation  relating  to 

the  annual  instalment  and  interest  upon  the  loan, 

as  described  in  Chapter  XXXV. 

The  Equation  of  the  Incidence  of  Taxation.  Chart  III. 

Showing  the  difference  between  the  total  annual  loan  charges 
in  respect  of  the  sinking  fund  instalment  and  interest  upon 
the  loan,  during  each  year  of  the  original  and  equated 
periods :  — 

(B).  Under  the  equated  method  as  generally  adopted, 
in  which  the  charge  is  spread  equally  over  the 
period. 

(D).  Under  the  method  described  in  Chapters  XXXIV. 
and  XXXV.,  in  which  the  revenue  or  rate  account 
is  charged  with  annual  sums  based  upon  the  life 
of  the  asset,  and  proportionate,  year  by  year,  to 
the  annual  charges  before  equation. 


THE    INCIDENCE    OF    TAXATION  451 

These  charts  show  in  graphic  form  :  — 

In  Chart  I,  the  total  annual  loan  charges  under  each 
method,  during  each  period  of  equal  incidence.  These  annual 
charges  are  divided  as  between  the  interest  upon  the  loan  which 
is  shown  in  the  lower  part  of  the  diagram,  and  the  annual 
instalment  which  is  shown  above  it.  The  height  of  each 
column  represents  the  total  annual  loan  charges,  and  the  width 
of  the  columns  represents  the  number  of  years  in  the  periods 
of  equal  incidence.  This  chart  brings  out  clearly  the  compara- 
tively small  relief  to  the  earlier  years  of  the  equated  period  and 
the  large  increased  annual  burden  during  the  final  eight  years 
of  such  period. 

In  Chart  II,  the  total  annual  charges  under  each  method 
during  each  period  of  equal  incidence  are  further  compared, 
but  without  any  subdivision  as  between  the  interest  upon  the 
loan  and  the  annual  instalment.  The  broken  line  shows  the 
equal  annual  burden  under  the  generally  adopted  method  after 
equation.  The  thin  unbroken  line  shows  the  annual  burdens 
under  the  original  conditions  before  equation,  which  were 
based,  both  as  regards  instalment  and  interest,  upon  the  life 
of  the  asset.  The  thick  unbroken  line  shows  the  corresponding 
annual  charges  under  the  author's  method  of  equation.  It  will 
be  noticed  that  the  two  unbroken  lines  agree  very  closely  and 
differ  widely  from  the  broken  line  of  the  generally  adopted 
method. 

In  Chart  III,  the  total  annual  loan  charges  under  the 
author's  method  of  equation  are  taken  as  the  standard  or  zero, 
and  are  compared,  as  to  the  equated  period,  with  the  charges 
under  the  generally  adopted  method  after  equation,  and  as  to 
the  post  equated  period  with  the  charges  under  the  original 
conditions.  The  area  below  the  zero  line  represents,  in  the  case 
of  the  equated  period,  the  relief  afforded  by  the  generally 
adopted  method  after  equation  as  compared  with  the  author's 
method,  and  as  regards  the  post  equated  period,  the  absolute 
relief  afforded  by  the  equation  of  the  period  irrespective  of  the 
method  in  which  the  burden  is  distributed  over  the  equated 
period.  The  area  above  the  zero  line,  which  occurs  only  in  the 
final  8  years  of  the  equated  period,  represents  the  additional 
annual  burden  imposed  upon  this  period  under  the  generally 
adopted  method  as  compared  with  the  author's  method. 
The  actual  amounts  of  relief  and  overcharge  are  taken  from 
Table  XXXV.  F.,  and  relate  to  the  loan  of  £56,000  used  to 
illustrate    the    problem,    consequently    any    comparison    based 


452    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

solely  upon  that  table  must  be  made  witb  tliis  actual  loan  in 
mind.  In  order  tberefore  to  show  tbe  results  in  a  form  wbicb 
will  be  readily  appreciated,  tbe  above  differences  bave  eacb  been 
expressed,  in  the  chart,  in  terms  of  an  annual  rate.  The  basis 
upon  which  this  has  been  done  is  a  statement  by  a  witness 
before  one  of  the  Parliamentary  Committees  appointed  to 
enquire  into  such  questions,  who  proved  that  in  a  particular 
ease  the  immediate  effect  of  an  equation  of  the  period  was  to 
reduce  the  rates  by  3d.  in  the  £  upon  the  annual  value.  This 
reduction  was  of  course,  only  between  the  annual  instalments, 
before  and  after  equation,  because  during  the  earlier  years  of 
the  equated  period,  as  shown  by  Chart  I,  there  is  not  any 
change  in  the  amount  of  interest  payable,  no  part  of  the  loan 
having  then  been  repaid  by  the  maturing  of  the  sinking  fund 
for  the  shorter  period.  If,  however,  the  comparison  be  made 
between  the  amount  payable  after  equation  and  the  proper 
amount  which  should  have  been  payable  under  the  author's 
method  as  a  consequence  of  such  equation,  the  saving  would 
be  6'52  pence  in  the  pound  instead  of  3  pence,  and  in  the  chart 
the  relief  to  the  first  part  of  the  equated  period  of  5  years  has 
been  taken  at  that  figure.  Fpon  this  basis,  the  effect  of  an 
equation  of  the  period,  in  the  present  instance,  under  the 
method  generally  adopted,  is  to  relieve  the  annual  rate  accounts 
as  follows  :  — 

during  the  equnted  period  :  — • 

for  a  period  of  5  years  of  6"52  pence  in  the  £. 
for  a  period  of  10  years  of  3'20  pence  in  the  £. 

during  the  post  equated  period  :  — 

for  a  period  of  6  years  of  9' 76  pence  in  the  £. 
for  a  period  of  16  years  of  2' 79  pence  in  the  £. 

and  to   impose   an    additional   annual   burden    upon   the   final 
8  years  of  the  equated  period  of  11 'So  pence  in  the  £. 

The  above  method  of  adjusting  the  annual  incidence  of  the 
total  loan  burden  may,  and  undoubtedly  will,  appear  com- 
plicated when  compared  with  the  rough  and  ready  method 
now  adopted.  It  will  certainly  increase  the  labour  involved 
upon  the  equation  of  the  period  of  repayment  of  new  loans 
authorised  for  outlays  of  varying  natures  and  also  upon  the 
consolidation  of  existing  loans,  but  it  is  sound  in  principle  and 
carries   out   the    fundamental    law   of   locnl    finance,   that   the 


THE    INCIDENCE    OF    TAXATION  453 

present  generation  shall  bear  at  least  its  due  burden  and  not 
transfer  it  to  future  years.  It  is  very  tempting  to  local  leaders 
of  finance  to  pose  as  tbe  benefactors  of  the  present  ratepayers 
by  adopting  a  method,  having  a  high-sounding  title,  which  has 
the  immediate  effect  of  reducing  the  present  burden  at  the 
expense  of  the  future ;  but  it  ought  to  be  recognised,  that  where 
the  only  means  of  paying  for  municipal  works  is  by  annual 
contributions  out  of  revenue  or  rate,  to  be  spread  over  a 
prescribed  period  of  years  fixed  after  very  careful  enquiry  as 
to  the  life  of  the  asset,  any  reduction  in  the  period  of  repay- 
ment, due  solely  to  causes  of  a  purely  financial  nature, 
cannot  possibly  equitably  reduce  the  annual  burden  but 
must  inevitably  increase  it.  Any  departure  from  this  principle 
is  a  violation  of  the  recognised  canons  of  local  government. 


A. 
B. 
C. 
D. 


under  original  conditions — before  equation. 

under  equated  method,  as  generally  adopted. 

Author's  equated  method  (annual  instalment  only). 

Author's  equated  method  (annual  instalment,  and  interest  on  the  loan) 


A. 
B. 
C. 
D. 


under  original  conditions — before  equation. 

under  equated  method,  as  generally  adopted. 

Author's  equated  method  (annual  instalment  only). 

Author's  equated  method  (annual  instalment,  and  interest  on  the  loan). 


A.  

B 

D 


under  original  conditions — before  equation. 

under  equated  method,  as  generally  adopted. 

Author's  equated  method,  (annual  instalment,  and  interest  on  the  loan 


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Appendix. 
Calculations  Referred  to  in  the  Text. 


Note.     Tlie  number  of  the  calculation  refers  to  tlie  chapter  to 
which  it  relates. 

The  detailed  working  of  the  method  (A)  by  formula  is  not 
included  in  any  of  the  following  calculations,  but  may  be 
found  by  referring  to  the  examples  given  in  the  text,  namely  : 

Form  No.  1.     Amount  of  one  pound.  No.  (lY)  3. 

2.  Present  value  of  one  pound.  (Y)  2. 

3.  Amount  of  one  pound  per  annum.  (YI)  2. 
3x.  Sinking  fund  instalment.                               (^^^)  1- 

4.  Present  value  of  one  pound  per  annum.       (YII)  2. 

5.  Annuity  which  one  pound  will  purchase.    (YIII)  2, 

Full  instructions  as  to  the  use  of  the  author's  standard 
calculation  forms  are  given  in  Chapter  X. 


462         REPAYMENT   OF   LOCAL   AND    OTHER    LOANS 


Standard  Form,  No.  5. 


No.  (XV)  3. 


To  find  the  additional  sinking  fund  instalment  to  be  set  aside 
during  the  unexpired  portion  of  tlie  repayment  period,  to 
compensate  for  a  deficiency  in  the  fund.  Table  Y. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
for  1'^  years  at  3|  per  cent.,  which  is  equivalent  to  a 
deficiency  of  £469'744  in  the  amount  now  in  the  fund. 

(C)     By  Thoman's  Table,  3^  per  cent.      Rule  3,  Chapter  YIII. 


Log.   Present  sum 

add  Log.  rt",  13  years 


469-744      2-6718612 
8-9870474 


11-6589086 


deduct  10 


1-6589086 


Required  annual  instalment 


£45-5941 


Standard  For  in.  To.  1. 


jNo.  (XT)  4. 


To  find  the  portion  of  original  loan  which  will  be  provided  by 
the  future  accumulation  of  the  present  investments  repre- 
senting the  fund.  Table  I. 

Required  the  amount  of  £9,463,  at  the  end  of  13  years, 
accumulated  at  3^  per  cent. 

(C)     By  Thoman's  Table,  3^  per  cent.  Rule  3,  Chapter  IT. 


Log.  Present  sum 

add  Log.  R^\  13  years... 


Required  future  amount 


9463      3-9760288 
.     ...       01942245 


41702533 


£14799-71 


APPENDIX 


Standard  Fortn,  No.  3. 


463 
No.  (XV)  5. 


To  find  the  portion  of  original  loan  which  will  be  provided  by 
the  accumulation  of  the  original  annual  instalments  to  be 
set  aside  during  the  unexpired  portion  of  the  redemption 
period.  Table  III. 

llequired  the  amount  of  an  annual  instalment  of  c£680"234,  to 
be  set  aside  and  accumulated  for  13  years  at  3^  per  cent. 

(C)     By  Thoman's  Table,  3i  per  cent.  Rule  3,  Chapter  VI. 


Log.    Annuity 680-234 

add  Log.  E^^  13  years,  +10       


deduct  Log.   a" 


Required  future  amount 


2-8326581 
10-1942245 

13-0268826 
8-9870474 

4-0398352 

£10960-62 


Standard  Form.  No.  1. 


No.  (XV)  6. 


To  find  the  portion  of  original  loan  which  will  be  unprovided 
if  the  present  deficiency  in  a  sinking  fund  be  allowed  to 
remain  uncorrected  during  the  remainder  of  the  redemption 
period.  Table  I. 

Required  the  amount  of  £469-744,  at  the  end  of  13  years, 
accumulated  at  3|  per  cent. 

(B)     By  Table  I,  13  years,  ^  per  cent.  Rule  2,  Chapter  IV. 


Log.  Amount  of  £1       

add  Log.  present  sum  ... 


1-56395      0-1942245 
469-744       2-6718612 


2-8660857 


Required  future  amount 


£734-659 


464         REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 
Standard  Form,  .To.  .3x.  No.  (XVI)  1. 

To  find  the  sinking  fund  instalment  required  to  provide  the 
amount  of  loan  represented  by  a  deficiency  in  the  fund. 

Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3^  per  cent.,  to  provide  £734'659  at  the  end  of  13  years. 

(C)     By  Thoman's  Table,  3i  per  cent.        Eule  3,  Chapter  XIII. 


Log.  Amount  of  loan  ... 

add  Log.  fl",  13  years. 


deduct  Log.  RN+iq 


Required  annual  instalment... 


734-659      2-8660857 
8-9870474 


11-8531331 
,..     10-1942245 

1-6589086 

£45-5941 


Standard  Form,  No.  3. 


Tso.  (XVI)  2. 


To  find   the   amount   of  loan   which   will  be   provided   by   the 

accumulation  of  the  augmented  annual  instalment. 

Table  III. 

Required  the  amount  which  should  stand  to  the  credit  of  a 
sinking  fund  representing  the  accumulation  of  an  annual 
instalment  of  £725-828  for  13  years  at  3|  per  cent. 

(]V)     By  Table  III,  13  years,  3i  per  cent.      Rule  2,  Chapter  YI. 

Log.  Amount  of  £1  per  annum  ...  16-11303       1-2071771 
ad^  Log.  annuity 725-828       2-8608337 


40680108 


Required  future  amount 


£11695-29 


APPENDIX 


Standard  Fonn,   No.  2. 


465 
No.  (XVI)  3. 


To  find  the  acciiiiiulated  anioimt  of  au  annual  instalment  to  be 
set  aside  for  a  limited  period  only ;  at  the  end  of  that  period 
the  amount  so  found  to  be  accumulated  for  a  further  pre- 
scribed period.      Method  by  "  step."  Table  II. 

Eequired  the  present  value  of  £7o4"659  due  at  the  end  of  8  years 
at  3^  per  cent. 


(C)     By  Thoman's  Table,  3i  per  cent. 


Eule  3,  Chapter  V. 


Log.    Future  sum 734-659       2-8660857 

deduct  Loo-.  RN^  8  years 0-1195228 


Required   present   value 


2-7465629 


£557-908 


Standard  Form,  No.  3x. 


No.  (XVI)  4. 


To  find  the  annual  instalment  which  will  amount  to  a  given  sum 
if  accumulated  for  a  prescribed  number  of  years.     Table  III . 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3i  per  cent,  to  provide  £557  908  at  the  end  of  5  years. 

(C)     By  Thoman's  Table,  3i  per  cent.        Rule  3,  Chapter  XIII. 


Log.   amount  of  loan     

add  Log.  a",  5  years    

deduct  Log.  RN+10  ... 
Required  annual  instalment... 


557-908      2-7405629 
9-3453372 


12-0919001 
100747017 

2-0171984 

£1040395 


466 


REPAYMENT   OF   LOCAL  AND   OTHER   LOANS 


Standard  Form,  No.  1 


No.  (XVI)  5. 


To  find  the  amount  of  loan  wliicli  will  be  provided  by  the  future 
accumulation  of  the  present  investments  representing  the 
fund.  Table  I. 

Required  the  amount  of  £9,46-3  at  the  end  of  5  years,  accumu- 
lated at  3^  per  cent. 

(C)     By  Thoman's  Table,  3i  per  cent.  Rule  3,  Chapter  lY. 


Log.  Present  sum 

add  Log.  R^,  5  years 


9463      3-9760288 
.     ...       00747017 


4-0507305 


Required  future  amount 


£11239-07 


Standard  Form,  No.  3. 


No.  (XVI)  6. 


To  find  the  amount  of  loan  which  will  bo  provided  by  the  future 
accumulation   of  the   augmented   annual   instalment. 

Table  III. 

Required  the  amount  of  an  annual  instalment  of  £784-273  to  be 
set  aside  and  accumulated  for  5  years  at  3^  per  cent, 

(C)     By  Thoman's  Table,  3^  per  cent.  Rule  3,  Chapter  YI. 

Log.  Annuity 784-273       2-8944673 

add  Log.  RN,  5  years,  +10 100747017 


deduct   Loff.  a" 


12-9691690 
9-3453372 

3-6238318 


Re(iuired  future  amount 


£4205-637 


APPENDIX 


467 


Standard  Form,  No.  1. 


No.  (XVI)  7. 


To   find   tlie   amount   of   loan   wliicli   will  be   provided   by   tlie 
accumulation  of  the  amount  in  the  fund.  Table  I. 

Required   the  anu)unt    of   £15444"71    at   the   end   of   8   years, 
accumulated  at  o\  per  cent. 

(13)     By  Table  I,  8  years,  S^  per  cent.  Rule  2,  Chapter  IV. 


Log.  Amount  of  £1       

add  Log.  present  sum  ., 


..    1-31681       0-1195228 
..15444-71      4-1887798 


4-3083026 


Required  future  amount 


£20337-74 


Standard  Form,  No.  3. 


No.  (XVI)  8. 


To  find  the   amount   of   loan  which  will  be   provided   by   the 
accumulation  of  the  original  annual  instalment.   Table  III. 

Required  the  amount  of  an  annual  instalment  of  £680-234  to  be 
set  aside  and  accumulated  for  8  years  at  3|  per  cent. 

(B)     By  Table  III,  8  years,  ^  per  cent.        Rule  2,  Chapter  VI. 

Log.  Amount  of  £1  per  annum  ...    905168       0-9567296 
add  Log.    annuity       680-234       2-8326581 

3-7893877 


Required  future  amount 


£6157-26 


468 


REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 


Standard  Form,  No.  3x. 


No.  (XVI)  9. 


To  find  the  future  annual  increment  to  be  accumulated  to 
provide  tlie  balance  of  loan  not  provided  by  tlie  present 
investments  representing  the  fund.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3^  per  cent,  to  provide  £17,032  at  the  end  of  13  years. 


(C)     By  Thoman's  Table,  3|  per  cent. 

Log.  Amount  of  loan    

add  Log.  a",  13  years  ... 


deduct  Log.  EN +  10 


Rule  3,  Chapter  XIII. 

.    17032         4-2312656 
8-9870474 


Required   annual  instalment 


13-2183130 
10-1942245 

3-0240885 

£1057-033 


Standard  Form.  No.  3,r 


No.  (XYT)  10. 


To  find  the  anniuil  sinking  fund  instalment  previoush'  set  aside 
in  error  to  provide  the  deficient  amount  now  in  the  fund. 

Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3^  per  cent,  to  provide  £9,463  at  the  end  of  12  years. 

(B)     By  Table  III,  12  years,  3i  percent.    Rule  2,  Chapter  XIII 

Log.  Amount  of  loan    9463'  3-9760288 

<:?eJwcf  Log.  amount  of  £1  per  ann.    146019       11044112 


2-8116176 


Re(|uired    aiinual    instalment 


£648-0636 


APPENDIX 


Standard  Form,  No.  5. 


469 
No.  (XVII)  1. 


To  find  tlie  amount  by  whicli  the  annual  sinking  fund  instalment 
may  be  reduced  in  consequence  of  a  payment  into  the  fund 
of  proceeds  of  sale  of  part  of  the  security  for  the  loan. 

Table  Y. 

Required  the  annuity  which  may  be  purchased  with  £4,560 
for  13  years  at  3^  per  cent. 

(B)     By  Table  V,  13  years,  3^  per  cent.     Eule  2,  Chapter  VIII. 

Log.  Annuity  £1  will  purchase  ...    0-097061     2-9870474 
a^^  Log.  present  sum 4560  3-6589648 


Required  annuity   ... 


2-6460122 


£442-6008 


Standard  Form,  JSo.  1. 


No.  (XVII)  2. 


To  find  the  portion  of  the  original  loan  which  will  be  provided 
by  the  future  accumulation  of  the  present  investments 
representing  the  fund.  Table  I. 

Required  the  amount  of  £9932*744  at  the  end  of  13  years, 
accumulated  at  3^  per  cent. 

(C)     By  Thoman's  Table,  3|  per  cent.  Rule  3,  Chapter  IV. 

Log.  Present  sum 9932-744       3-9970693 

aiZfZ  Log.  RN,  13  years 01942245 


41912938 


Required  future  amount 


£15534-375 


470         REPAYMENT   OF   LOCAL  AND   OTHER  LOANS 
Stcmdard  Form,,  No.  1.  No.  (XVII)  3. 

To  find  the  portion  of  the  original  loan  which  will  be  provided 
by  the  future  accumulation  of  the  proceeds  of  sale  of  assets 
paid  into  the  fund.  Table  I. 

Required  the  amount  of  £4,560  at  the  end  of  13  years,  accumu- 
lated at  3^  per  cent. 

(B)     By  Table  I,  13  years,  at  3^  per  cent.      Rule  2,  Chapter  IV. 

Log.  Amount  of  £1       1-66395       01942245 

a<Z^  Log.  present  sum 4560  3"6589648 


3-8531893 


Required  future  amount       £7131*64 


Standard  Forvi,  No.  3x.  No.  (XVII)  4. 

To  find  the  annual  instalment  which  will  provide  the  amount  of 
loan  represented  by  the  proceeds  of  sale  of  part  of  the 
security  paid  into  the  fund.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3|  per  cent,  to  provide  £7131-64  at  the  end  of  13  years. 

(C)     By  Thomun's  Table,  3i  per  cent.       Rule  3,  Chaper  XIII. 

Log.  Amount  of  loan 7131-64       38531893 

a^Z^/' Log.  a«,  13  years 8-9870474 


12-8402367 
cZe^act  Log.  RN+10 101942245 


2-6460122 


Required  annual  instalment £4426008 


APPENDIX  471 

Standard  Form,  No.  3.  No.  (XVII)  5. 

To  fiud  the  amount  of  loan  which  will  be  provided  by  the  future 
accumulation  of  the  reduced  annual  instalment  consequent 
upon  the  payment  into  the  fund  of  the  proceeds  of  sale  of 
part  of  the  security.  Table  III. 

Required  the  amount  of  an  annual  instalment  of  <£237'633  to 
be  set  aside  and  accumulated  for  13  years   at  3|  per  cent. 

(C)     By  Thoman's  Table,  3^  per  cent.  Rule  3,  Chapter  YI. 

Log.  Annuity         237-633       2-3759068 

add  Log.  RN,  13  years,  +10     101942245 


12-5701313 
deduct  Loff.  a" 8-9870474 


3-5830839 


Required  future  amount       £3823*987 


Standard  Form,  No.  3x.  No.  (XVII)  6. 

To  find  the  future  annual  increment  to  be  accumulated  to 
provide  the  balance  of  loan  not  provided  by  the  investments 
representing  the  fund.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3|  per  cent,  to  provide  £1200226  at  the  end  of  13  years, 

(B)     By  Table  III,  13  years, 3^ percent.     Rule  2,  Chapter  XIII. 

Log.  Amount  of  loan    1200226     4-0792630 

deduct  Log.  amount  of  £1  per  aun.   16-11303     1-2071771 


2-8720859 


Required  annual  instalment £744-879 


472  REPAYMENT   OF   LOCAL   AND    OTHER   LOANvS 

Standard  Form,  Xo.  3it.  No.  (XVIII)  1. 

To  fiud  the  aniouut  by  which  the  original  annual  instalment 
may  be  reduced  in  consequence  of  the  withdrawal  of  part 
of  the  loan  from  the  operation  of  the  fund  by  reason  of  its 
conversion  into  ordinary  share  capital  or  stock.     Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  '6\  per  cent,  to  provide  £5,000  at  the  end  of  13  years. 

(C)     By  Thoman's  Table,  o\  per  cent.        Rule  3,  Chapter  XIII. 

Log.  Amount  of  loan    5000  3-6989700 

add  Log.  a'S  13  years         8-9870474 

12-6860174 
Je^m-t  Log.  RN+io 10-1942245 

2-4917929 


Required  annual  instalment       £310-308 


Standard  Form,  No.  3a',.  >'«•  (XVIII)  2. 

To  find  the  amended  annual  instalment  which  will  provide  the 
balance  of  loan  not  provided  by  the  future  accumulation 
of  the  present  investments  and  the  original  annual  instal- 
ment, after  withdrawal  of  part  of  the  loan  from  the  opera- 
tion of  the  fund.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  ^  i)er  cent,  to  provide  £6695-30  at  the  end  of  13  years. 

(13)     By  Table  III,  13  years  3i  per  cent.    Rule  2,  Chapter  XIII. 

Log.  Amount  of  loan 669530       3-8257700 

deduct  Log.  amount  of  £1  per  ann.  1611303       1*2071771 

2-6185929 


Required  annual  instalment £415-520 


APPENDIX  473 

Standard  Form,  Xo.  3x.  No.  (XVIII)  3. 

To  find  tlie  future  annual  increment  to  be  accumulated  to 
provide  the  balance  of  loan  not  provided  by  the  present 
investments  representing'  the  fund.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  o\  per  cent,  to  provide  i^l2,0o2  at  the  end  of  13  years. 

(C)     By  Thoman's  Table,  ^  per  cent.       Rule  3,  Chapter  XIII. 

Lo^.  Amount  of  loan 12032  4-0803378 

aJ(Z  Log.  (/«,  13  years 89870474 


13-0673852 
^(■J,/rf  Log.  RN  +  10 101942245 


2-8731607 


Required  annual  instalment £746-725 


Standard  Form,  No.  3.  ^'o-  (XVIII)  4. 

To  find  the  amount  which  should  stand  to  the  credit  of  the 
sinking  fund,  lable  111. 

Required  the  amount  of  an  annual  instalment  of  £7,500  to  be 
set  aside  and  accumulated  for  7  years  at  3  per  cent. 

(B)     By  Table  III,  7  years,  3  per  cent.         Rule  2,  Chapter  YI. 

Log.  Amount  of  £1  per  annum  ...    7-6625         0-8843684 
a(^(Z  Loff.  annuity 7500  3-8750613 


4-7594297 


Required  future  amount       £57468-48 


474  REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 


Standard  Form-,  No.  Sx.  >'o.  (XVIII)  5. 

To  find  the  annual  instalment  for  an  even  number  of  years 
which  ajDproximates  to  the  instalment  of  specified  amount 
not  found  by  calculation.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3  per  cent,  to  provide  £150,000  at  the  end  of  16  years. 

(B)     By  Table  III,  16  years,  3  per  cent.     Rule  2,  Chapter  XIII. 

Log.  Amount  of  loan 150,000       51760913 

deduct  Log.  amount  of  £1  per  ann.    20' 1569       13044233 


3-8716680 


Required  annual  instalment £74416285 


Standard  Form,  No.  3.  No.  (XVUI)  6. 

To  find  the  amount  of  loan  which  will  be  provided  by  the 
instalment  of  stated  amount  at  the  end  of  the  approximate 
period  of  even  years.  Table  III. 

Required  the  amount  of  an  annual  instalment  of  £7,500  to  be 
set  aside  and  accumulated  for  16  years  at  3  per  cent. 

(C)     By  Thoman's  Table,  3  per  cent.  Rule  3,  Chapter  VI. 

Log.  Annuity         7500  38750613 

add  Log.  RN,  16  years,  +10       10-2053956 


14-0804569 
deduct  Log.  r/«    8-9009723 


5-1794846 


Required  future  amount       £15117659 


APPENDIX 


475 


Standard  Form,  No.  3. 


No.  (XVIII)  7. 


To  find  the  portion  of  the  original  loan,  being  the  accumulation 
of  an  intentional  error  in  the  sinking-  fund  instalment 
assumed  in  the  adjustment.  Table  III. 

Required  the  amount  of  £58-3715  per  annum  for  16  years  at 
3  per  cent. 

(B)     By  Table  III,  16  years,  3  per  cent.        Rule  2,  Chapter  VI. 

Log.  Amount  of  £1  per  annum    ...    201569       1-3044233 
a^Z^  Log.  annuity 58-3715       r7662008 


30706241 


Required  future  amount 


£1176-58 


Standard,  Form,  No.  3x. 


No.  (XVIII)  8. 


To  find  the  amount  by  Avhich  the  original  annual  instalment 
may  be  reduced  in  consequence  of  the  withdrawal  of  part 
of  the  loan  from  the  operation  of  the  sinking  fund  by 
reason  of  its  conversion  into  ordinary  share  capital  or 
stock.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3  per  cent,  to  provide  £45,000  at  the  end  of  9  years. 

(B)     By  Table  III,  9  years,  3  per  cent.       Rule  2,  Chapter  XIII. 

Log.  Amount  of  loan    45000       4-6532125 

^eiZwci  Log.  amount  of  £1  per  ami.    101591       1-0068555 


3-6463570 


Required  annual  instalment. 


£4429-523 


476 


REPAYMENT   OF   LOCAL   AND    OTHER   LOANS 


Staiulard  Form,  Xo.  3. 


No.  (XVIII)  9. 


To  find  the  aiiiuiiut  wliieli   sliuuld  stand  to  the   credit   of  the 
fund.  Table  III. 

Required  the  amount  of  an  annual  instalment  of  £7441'6285  to 
be  set  aside  and  accumulated  for  7  years  at  -3  per  cent. 


(C)     By  Thoman's  Table,  3  per  cent. 

Log.  Annuity        

add  Log.  E.N^  7  years,  + 10 


deduct  Log.  a" 


E-ule  o,  Chapter  YI. 

7441-6285      3-8716680 
100898606 


Required  future  amount 


13-9615286 
9-2054922 

4-7560364 
£57021-21 


Standard  Form,  No.  5. 


No.  (XVIII)  10. 


To  find  the  amount  by  which  the  annual  instalment  may  be 
reduced  in  consequence  of  a  surplus  in  the  fund.      Table  Y. 

Required  the  annuity  which  may  be  purchased  with  £44727  for 
9  years  at  3  per  cent. 

(C)     By  Thoman's  Table,  3  per  cent.  Rule  3,  Chapter  YIII. 


Log.   Present  sum 

add.  Log.  a",  9  years 


447-27       2-6505698 
9-1086795 


11-7592493 


deduct  10     ... 
Required  annuity  ... 


1-7592493 


£57-4446 


APPENDIX 


477 


Standard  Form,  No.  1. 


ISo.  (XVIII)  11. 


To  find  tlie  amount  of  loan  wliirli  will  be  provided  by  tbe  fiiture 
accumulation  of  the  present  investments  of  the  amount 
in  the  fund.  Table  I. 

Eequired  the  amount  of  £57468-48  at  the  end  of  9  years, 
accumulated  at  3  per  cent. 

(C)     By  Thoman's  Table,  3  per  cent.  Rule  3,  Cliapter  lY. 

Log.  Present  sum 59468-48       4-7594297 

rtfZfZ  Log.  RN    9  years 0-1155350 


4-8749647 


Required  future  amount 


£74983-335 


Standard  Form,  No.  3. 


No.  (XVIII)  12. 


To  find  the  amount  of  loan  which  will  be  provided  by  the  future 
accumulation  of  the  reduced  annual  instalment  in  conse- 
quence of  the  withdrawal  of  part  of  the  loan  from  the 
operation  of  the  fund.  Table  III. 

Required  the  amount  of  an  annual  instalment  of  £295466  to  be 
set  aside  and  accumulated  for  9  years  at  3  per  cent. 

(B)     By  Table  III,  9  years,  3  per  cent.        Rule  2,  Chapter  YI. 

Log.  Amount  of  £1  per  annum  ...    101591        1-0068555 
«^^  Log.  annuity 2954-66       3-4705075 

4-4773630 


Required  future  amount 


£30016-70 


478    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Standard  Form,  No.  3. 


No.  (XVIII)  13. 


To  find  the  amount  of  loan  represented  by  the  adjustment  to  be 
made  in  the  annual  instalment  in  consequence  of  an  inten- 
tional error  introduced  for  purpose  of  calculation.  Tablelll. 

Required  the  amount  of  an  annual  instalment  of  £40"215  to 
be  set  aside  and  accumulated  for  9  years  at  3  per  cent. 

(C)     By  Thoman's  Table,  3  per  cent.  Rule  3,  Chapter  YI. 

Log.  Annuity  40-215       r6043881 

«^cZ  Log.  RN,  9years,  +10 10-1155350 


deduct  Log.  a" 


Required  future  amount 


11-7199231 
91086795 

2-6112436 

£408-549 


Standard  Form,  No.  4. 


No.  (XVIII)  14. 


To  find  the  present  amount  to  be  paid  into  the  fund  to 
compensate  for  the  intentional  error  introduced  for  purposes 
of  calculation.  Table  IV. 

Required  the  present  value  of  an  annuity  of  £40215  for  9  years 
at  3  per  cent. 


(C)     By  Thoman's  Table,  3  per  cent. 
Log.  Annuity  


Rule  3,  Chapter  y II. 
40-215       1-6043881 


add  10     

deduct  Log.  a'*,  9  years    ., 


Required  present  value 


11-6043881 
9-1086795 

2-4957086 

£313-118 


APPENDIX 


479 


Standard  Form,  No.  3x. 


No.  (XVIII)  15. 


To  find  the  future  annual  increment  to  be  accumulated  to 
provide  tlie  balance  of  loan  not  provided  for  by  the  present 
investments  representing  the  fund.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3  per  cent,  to  provide  £47531-52  at  the  end  of  9  years. 

(C)     By  Thoman's  Table,  3  per  cent.  Eule  3,  Chapter  XIII. 


Log.  Amount  of  loan  ... 
add.  Log.  a",  9  years 


deduct  Log.  EN +10  .. 


Required   annual   instalment 


47531-52      4-6769818 
9-1086795 


13-7856613 
10-1155350 

3-6701263 

£4678-71 


Standard  Form,  No.  3. 


No.  (XIX)  1. 


To  find  the  portion  of  original  loan  which  will  be  provided  by 
the  future  accumulation  of  the  annual  income  from  the 
present  investments.  Table  111. 

Required  the  amount  of  an  annual  income  of  £347  648  to  be 
added  to  the  sinking  fund  and  accumulated  for  13  years 
at  3  per  cent. 

(B)     By  Table  III,  13  years,  3  per  cent.      Rule  2,  Chapter  VI. 

Log.  Amount  of  £1  per  annum  ...    15-6178       1-1936196 
a^cZ  Log.  annuity 347648       2-5411397 

3-7347593 


Required  future  amount 


£5429-494 


48o         REPAYMENT   OF   LOCAL   AND   OTHER   LOANS 


Standard  Form,  No.  3. 


>o.  (XIX)  2. 


To  find  the  portion  of  tlie  originai  loan  wliicli  will  be  provided 
by  the  future  accumulation  of  the  original  instalment. 

Table  III. 

Required  the  amount  of  an  annual  instalment  of  £680'234  to  be 
set  aside  and  accumulated  for  l-J  years  at  o  per  cent. 


(C)     By  Thoman'sTable,  3per  cent. 


Rule  3, Chapter  VI. 


Log.  Annuity         680-234      2-8826581 

arZ^Z  Log.  RN,  13  years,  + 10       10-1668839 


deduct  Log.   a'^   .. 


Required  future  amount 


12-9995420 
8-9732643 

4-0262777 

£10623-75 


Standard  Form,  No.  3x. 


^0.  (XIX)  3. 


To  find  the  additional  sinking  fund  instalment  to  compensate 
for  a  reduction  in  the  rate  of  accumulation.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3  per  cent,  to  provide  £509- 02  at  the  end  of  13  years. 

(B)     By  Table  III,  13  years,  :{  per  cent.    Rule  2,  (liapter  XIII. 


Loff.  Amount  of   h)an   ... 


509-02       2-7067348 


deduct  Log.  amount  of  £1  per  ami.    156178       1-1936196 


1-5131152 


Required   annual    instalment 


£32-5923 


APPENDIX 


481 


Standard  Form,  No.  3. 


No.  (XIX)  4. 


To  find  tlie  portion  of  the  orio^inal  loan  which  will  be  provided 
by  the  aceumulation  of  the  amended  annual  increment. 

Table  III. 

Required  the  amount  of  an  annual  increment  of  £1060'474  to 
be  added  to  the  sinking-  fund  and  accumulated  for  1^  years 
at  3  per  cent. 

(B)     By  Table  III,  13  years,  3  per  cent.        Rule  2,  Chapter  YI. 

Log.  Amount  of  £1  per  annum  ...  15"617T9       1"1936196 
add  Log.  annuity 1060-474       3-0255000 


4-2191196 


Required  future  amount 


£16562-26 


Standard  Form,  No.  3x. 


No.  (XIX)  5. 


To  find  the  amended  annual  instalment  to  repay  the  balance  of 
loan  at  the  end  of  the  period  of  repayment.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3  per  cent,  to  provide  £1656226  at  the  end  of  13  years, 

(C)     By  Thoman's  Table,  3  per  cent.       Rule  3,  Chapter  XIII. 

Log.  Amount  of  loan 16562-26       4-2191196 

flfZfZ  Log.  ««,  13  years 8-9732643 


deduct  Log.  RN  +  10  . 


Required  annual  instalment 


13-1923839 
10-1668839 

30255000 

£1060-474 


A  G 


482    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Standard  Forw,  No.  3.  ^o-  (XX)  1. 

To  find  the  amount  of  loan  wliicli  will  be  provided  by  the  futiire 
accumulation  of  the  income  from  the  present  investments 
representing  the  fund.  Table  III. 

Required  the  amount  of  an  annual  income  of  £297'984  to  be 
added  to  the  sinkino-  fund  and  accumulated  for  13  years 
at  3  per  cent. 

(C)     By  Thoman's  Table,  3  per  cent.         Rule  3,     Chapter  YI. 

Loc..  Annuity  297-984       2-4741929 


add  Lo^.  RN,  13  years,  +10 


10-1668839 


12-6410768 
deduct  Loo-,  ^n 89732643 


Required  future  amount 


3-6678125 
£4653-85 


Standard  Form  No.  3x.  ^»-  (XX)  2. 

To  find  the  additional  annual  instalment  to  provide  the  amount 
of  loan  unprovided  for  owing  to  a  reduction  in  the  rate  of 
income  from  the  present  investments.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3  per  cent,  to  provide  £77564  at  the  end  of  13  j-ears. 

(B)     By  Table  III,  13  years,  3  per  cent.    Rule  2,  Chapter  XIII. 

Log.  Amount  of  Loan 775-64        2-8896602 

deduct  Log.  amount  of  £1  per  ann.    1561779     1-1936196 

1-6960406 


Required  annual  instalment £49-664 


APPENDIX 


483 


Standard  Form,  No.  3. 


No.  (XX)  3. 


To  find  the  amount  of  loan  wliirli  will  bo  provided  by  the 
accumulation  of  the  future  annual  increment.       Table  III. 

Required  the  amount  of  an  anniml  increment  of  £1060'4T4  to 
be  added  to  the  sinking  fund  and  accumulated  for  13  years 
at  3  per  cent. 

(B)     By  Table  III,  13  years,  3  per  cent.        Eule  2,  Chapter  YI. 

Log.  Amount  of  £1  per  anniini  ...    156178       1-1936196 
^«W  Loo-,  annuity 1060474     3-0255000 


Required  future  amount 


4-2191196 
£16562-26 


Standard  Form,  No.  3x. 


No.  (XX)  4. 


To  find  the  amended  annual  instalment  consequent  upon  a 
variation  in  the  rate  of  income  upon  the  present  invest- 
ments, but  without  any  variation  in  the  rate  of  accumula- 
tion. Table  III. 

Required  the  annual  increment  to  be  added  to  the  sinking 
fund  and  accumulated  at  3  per  cent,  to  provide  £16562-26 
at  the  end  of  13  years. 

(C)     V>y  Thoman's  Table,  3  per  cent.        Rule  3,  Chapter  XIII. 

Log.  Amount  of  loan 1656226       4-2191196 

rtrffZ  Lop-.  r/«    13  years 8-9732643 


deduct  Lost.  R^+IO 


Required  annual  instalment. 


13-1923839 
,.     10-1668839 

3-0255000 

..      £1060-474 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Standard  Form,  No.  3.  No.  (XXI)  1. 

To  find  tlie  amount  of  loan  wliicli  will  be  provided  by  the 
accumulation  of  the  annual  income  from  tbe  present  invest- 
ments under  tbe  altered  conditions.  Table  III. 

Required  the  amount  of  an  annual  income  of  <£297'984  to  be 
added  to  tbe  sinkinfr  fund  and  accumulated  for  13  years  at 
2^  per  cent. 

(C)     By  Thoman's  Table,  2^  per  cent.  Rule  3,  Cbapter  YI. 

Log.  Annuity 297-984       2-4741929 

add  Log.  RN^  i:>  years,  +10     101394103 

12-6136032 
deduct  Log.  «« 8-9592717 


3-6543315 


Required  future  amount       £4511-6094 


Standard  Form,   No.   3.  No.    (XXI)   2. 

To  find  tbe  amount  of  loan  wbicb  will  be  provided  by  tbe 
accumulation  of  tbe  original  annual  instalment  under  tbe 
altered  conditions.  Table  III. 

Required  tbe  amount  of  an  annual  instalment  of  £680234  to  be 
set  aside  and  accumulated  for  13  years  at  2^  per  cent. 

(B)     By  Table  III,  13  years,  2^  per  cent.    Rule  2,  Cbapter  VI. 

Log.  Amount  of  £1  per  annum  ...    1514044     1*1801386 
«J^I  Log.  annuity 680234       28326581 

40127967 


Required  future  amount       £10299038 


APPENDIX  485 

Standard  Form,   No.   3.  No.    (XXI)    3. 

To  find  tlie  amount  of  loan  wliicli  will  be  provided  by  the 
accumulation  of  the  additional  annual  instalment  under 
the  altered  conditions.  Table  III. 

Required  the  amount  of  an  annual  instalment  of  £32"592  to  be 
set  aside  and  accumulated  for  13  years  at  2\  per  cent. 

(C)     By  Thoman's  Table,  2\  per  cent.  Rule  3,  Chapter  VI. 

Log.  Annuity  32592         1-5131152 

add  Log.  RN,  13  years,  +10     101394103 

11-6525255 
deduct  Log.  a'' 89592717 


2-6932538 


Required  future  amount       £493-462 


Standard  Form,  No.  3x.  No.  (XXI)  4. 

To  find  the  annual  instalment  required  to  provide  the  balance 
of  loan  which  will  be  unprovided  for  owing  to  a  reduction 
in  the  rate  of  accumulation,  etc.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  2i  per  cent,  to  provide  £1258-15  at  the  end  of  13  years. 

(B)     By  Table  III,  13  years,  2^  per  cent.    Rule  2,  Chapter  XIII. 

Log.  Amount  of  loan 125815       30997325 

deduct  Log.  amount  of  £1  per  ann.    15-14044     1-1801386 

1-9195939 


Required  annual  instalment £83-0986 


486 


REPAYMENT    OF    LOCAL   AND    OTHER   LOANvS 


Standard   Fortn,    No.    3. 


IVo.    (XXI)    5. 


To  prove  tliat  the  amended  annual  increment  as  ascertained 
will  complete  tlie  linal  repayment  of  tlie  loan  under  the 
altered  conditions.  Table  III. 

llequired  the  amount  of  an  annual  increment  of  £1093'909  to  he 
added  to  the  sinking  fund  and  accumulated  for  V-^  years  at 
2\  per  cent. 

(B)     By  Table  III,  I'i  years,  2^  per  cent.       Eule  2,  Chapter  VI. 

Log.  Amount  of  £1  per  annum  ...    15-14044     1-1801386 
«^'/ Log.  annuity 1093-909     3-0389812 


4-2191198 


llequired  future  amount 


£16562-26 


Standard  Form,  No.  1. 


]No.  (XXIV)  1. 


To  find  the  anunint  of  loan  which  will  be  provided  l)y  the  future 
accumulation  of  the  present  investments  representing  the 
fund.  Table  I. 

llequired  the  amount  of  £9932-744  at  the  end  of  8  years, 
accumulated  at  3^  per  cent. 

[\\)     By  Table  I,  8  years,  3|  per  cent.  llule  2,  Chapter  I\'. 


Log.  Amount  of  £1       

add  Log.  present  sum  .. 


1-31681       0-1195228 
9932-744    3-9970693 


41165921 


llequired  future  amount 


£13079-53 


APPENDIX 


487 


Standard  Form,  No.  3. 


>o.  (XXIV)  2. 


To  find  the  amount  of  loan  whicli  will  be  provided  by  tbc  future 
accumulation  of  tbe  original  annual  instalment.     Table  III. 

Required  tbe  amount  of  an  annual  instalment  of  £680'234  to  be 
set  aside  and  accumulated  for  8  years  at  3|  per  cent. 

(C)     By  Tboman's  Table,  3i  per  cent.  Rule  3,  Chapter  YI. 

Log.  Annuity  680-234       2-8326581 

«^rf  Log.  RN  8  years,  +  10 101195228 


deduct  Log.  ci^      ...     ... 

Required  future  amount 


12-9521809 
...       9-1627932 

3-7893877 

...    £6157-2614 


Standard  Form,  No.  3x. 


No.  (XXIV)  3. 


To  find  tbe  additional  annual  instalment  to  be  set  aside  and 
added  to  tbe  fund  to  compensate  for  tbe  reduction  in  tbe 
redemption  period.  Table  111. 

Required  tbe  annual  instalment  to  be  set  aside  and  accumulated 
at  3i  per  cent,  to  provide  £725821  at  tbe  end  of  8  years. 

(C)     By  Tboman's  Table,  3^  per  cent.        Rule  3,  Chapter  XIII. 


Log.  Amount  of  loan   ... 
add  Log.  fl",  8  years 


7258-21      3-8608294 
9-1627932 


deduct  Log.  RN+10 

Required  annual  instalment... 


130236226 
101195228 

2-9040998 

£801-8624 


488 


REPAYMENT   OF    LOCAL   AND    OTHER   LOANS 


Standard  Form,  No.  3. 


No.  (XXIY)  4. 


To  find  the  amount  of  loan  whicli  will  be  provided  by  tlie  future 
accumulation  of  the  present  annual  income  from  invest- 
ments for  the  unexpired  portion  of  the  repayment  period. 

Table  III. 

Required  the  amount  of  an  annual  income  of  ^347'648  to  be 
added  to  the  sinking  fund  and  accumulated  for  13  years  at 
3|  per  cent. 

(B)     By  Table  III,  13  years,  3^  per  cent.     Rule  2,  Chapter  YI. 

Log.  Amount  of  <£1  per  annum  ...    1611303     1-2071771 
flfZfZ  Log.  annuity 347'648       2-5411397 


Required  future  amount 


3-7483168 
£5601-66 


Standard-  Form,  No.  3. 


No.  (XXIV)  5. 


To  find  the  amount  of  loan  which  will  be  provided  by  the  future 
accumulation  of  the  present  annual  income  from  invest- 
ments at  the  end  of  the  substituted  period  of  repayment. 

fable  III. 

Required  the  amount  of  an  annual  income  of  £347-648  to  be 
added  to  the  sinking  fund  and  accumulated  for  8  years  at 
3^  per  cent. 

(Cj     By  Thoman's  Table,  3^  per  cent.         Rule  3,  Chapter  A'l. 


Log.  Annuity  

add  Log.  1{N,  8years,  +10 

deduct   L()<r.   a^   ... 


Recjuired  future  amount 


347-648      2-5411397 
10-1195228 


12-6606625 
9-1627932 

3-4978693 

£3146-81 


APPENDIX 


489 


Standard  Form,  No.  3. 


No.  (XXIY)  6. 


To  find  tlie  amount  of  loan  which  will  be  provided  at  the  end 
of  the  substituted  period  of  repayment  by  the  accumulation 
of  the  amended  annual  increment.  Table  III. 

Required  the  amount  of  an  annual  increment  of  £1829" 744  to 
be  added  to  the  sinking  fund  and  accumulated  for  8  years 
at  3|  per  cent. 

(B)     By  Table  III,  8  years,  3i  per  cent.       Rule  2,  Chapter  VI. 

Log.  Amount  of  £1  per  annum  ...    9-05169       0-9567296 
acZtZ  Log.  annuity 1829744     ;j-2623904 


Required  future  amount 


4-2191200 
£16562-26 


Standard  Form,  No.  3. 


No.  (XXVI)  1. 


To  find  the  amount  of  loan  which  will  be  provided  by  the  future 
accumulation  of  the  present  annual  instalment  at  the  end 
of  the  substituted  period  of  repayment.  Table  III. 

Required  the  amount  of  an  annual  instalment  of  £712-826  to  be 
set  aside  and  accumulated  for  8  years  at  o^  per  cent, 

(C)     By  Thoman's  Table,  o^  per  cent..        Rule  3,  Chapter  VI. 


Log.  Annuity  

add  Log.  RN,  8  years,  +  10  . 

deduct   Loo;,   a"   ... 


Required  future  amount 


712-826      2-8529836 
10-1195228 


12-9725064 
9-1627932 

3-8097132 

£6452-28 


490    REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 

Standard  Form,  No.  3x.  No.  (XXVI)  2. 

To  find  the  additioual  annual  instalment  to  be  set  aside  and 
added  to  the  fund  in  consequence  of  a  reduction  in  the 
period  of  repayment  accompanied  by  an  increase  in  the 
rate  of  accumulation.  Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3|  per  cent,  to  provide  £696-j-19  at  the  end  of  8  years. 

(B)     By  Table  III,  8  years,  ^  per  cent.     Kule  2,  Chapter  XIII. 

Log.  Amount  of  loan      6963-19       3-8428082 

deduct  Log.  amount  of  £l  per  ann.    9-06169       0-9567296 


2-8860786 


llequired  annual  instalment £769-270 


Standard  Form,  No.  3.  No.  (XXVI)  3. 

To  find  the  amount  of  loan  which  will  be  provided  by  the 
accumulation  of  the  present  annual  increment.     Table  III. 

Kcfjuired  the  amount  of  an  annual  increment  of  £1060-474  to  be 
added  to  the  sinking  fund  and  accumulated  for  8  years  at 
3  per  cent. 

(B)     ]Jy  Table  III,  8  years,  3  per  cent.  Kulc  2,  Chapter  VI 

Log.  Amount  of  £1  per  annum  ...    889234       09490159 
add  Log.  annuity         1060474     3-0255000 


3-9745159 


llequired  future  amount       £943009 


APPENDIX 


491 


Standard  Form,  No.  3. 


No.  (XXVI)  4. 


To   find   the   amount  of   loan   wliieli   will   l>e   provided    by   tlie 
accumulation  of  the  amended  annual  instalment.  Table  III. 

Required  the  amount  of  an  annual  instalment  of  £816'232  to  be 
set  aside  and  accumulated  for  8  years  at  3  per  cent. 

(C)     By  Thoman's  Table,  3  per  cent.  Rule  3,  Chapter  VI. 


Log.  Annuity  

add  Log.  RN,  8  years,  +10 

deduct  Loa;.   a'^ 


Required  future  amount 


816-232      2-9118125 
101026978 


13-0145103 
91536819 

3-8608284 

£7258-19 


Standard  Form,  No.  3x. 


No.  (XXVI)  .5. 


To  find  the  amount  by  which  the  annual  instalment  may  be 
reduced  in  consequence  of  a  surplus  of  loan  which  will  be 
provided  by  an  excessive  annual  instalment.         Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3  per  cent,  to  provide  £126-04  at  the  end  of  8  years. 

(Cj      By  Thoman's  Table,  3  per  cent.  Rule  3,  Chapter  XIII. 


Log.  Amount  of  loan    126-04 

flc/^  Log.  a",  8  years     

deduct  Loff.  RN+10 


Required  annual  instalment... 


21005084 
9-1536819 


11-2541903 
101026978 


11514925 
£14-174 


492 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Standard  Form,  iVo.  3. 


No.  (XXVII)  1. 


To  find  the  portion  of  original  loan  wliicL.  will  be  provided  by 
the  future  accumulation  of  the  varying  annual  income  from 
the  present  investments,  being  the  first  stage  in  the  method 
by  step  Table  III. 

Required  the  amount  of  an  annual  income  of  £-)4T'648  to  be 
added  to  the  sinking  fund  and  accumulated  for  8  years  at 
3  per  cent. 

(C)     By  Thoman's  Table,  3  per  cent.  Rule  3,  Chapter  YI. 


Log.  Annuity 

add  Log.  RN,   8  years,  +10 

deduct  Log.  a^ 

Required  future  amount 


347-648      2-5411397 
10-1026978 


12-6438375 
91536819 

3-4901556 

£3091-403 


Standard  Form,  3  o.  1. 


No.  (XXVII)  2. 


To  find  the  accumulated  amount,  at  the  end  of  the  unexpired 
period,  of  the  amount  found  in  the  previous  calculation, 
being  the  second  stage  in  the  method  by  step.  Table  I. 

Required  the  amount  of  £3091-403  at  the  end  of  5  years 
accumulated  at  3  per  cent. 

(C)     By  Thoman's  Tabh',  .'*,  i)er  cent.  Rule  3,  Chapter  lY. 


Log.   Pieseiit  sum 

add  Log.  R^,  5  years  .. 


Required  future  amount 


3091-403       3-4901556 
00641861 


3-5543417 

£3583-783 


APPENDIX  493 


Standard  Form,  No.  3.  IVo.  (XXVIT)  3. 

To  find  the  portion  of  original  loan  which  will  he  provided  hy 
the  futnre  accumulation  of  the  reduced  annual  income  from 
the  present  investments  during  the  second  part  of  the 
unexpired  repayment  period.  Table  III. 

Eeqiiired  the  amount  of  an  annual  income  of  £297"984  to  he 
added  to  the  sinking  fund  and  accumulated  for  5  years  at 
3  per  cent. 

(C)     By  Thoman's  Table,  3  per  cent.  Rule  3,  Chapter  VI. 

Log.  Annuity 297-984       2-4741929 

acU  Log.  RN,  5  years,  +10       10-0641861 


12-5383790 
deduct  Loff.  a« 9-3391623 


31992167 


Required  future  amount       £1582-037 


Standard  Form,  No.  Sx.  No.  (XXVII)  5. 

To  find  the  additional  annual  instalment  required  in  consequence 
of  a  reduction  in  the  rate  of  income  from  investments 
during  the  later  years  of  the  unexpired  period  of  repay- 
ment. Table  III. 

Required  the  annual  instalment  to  be  set  aside  and  accumulated 
at  3  per  cent,  to  provide  £26367  at  the  end  of  13  years. 

(B)     By  Table  III,  13  years,  3  per  cent.    Rule  2,  Chapter  XIII. 

Log.  Amou2it  of  loan 26367         2-4210607 

deduct  Log.  amount  of  £1  per  ann.    156178       1-1936196 

1-2274411 


Required  annual  instalment £16-8827 


\ 


494 


REPAYMENT  OF  LOCAL  AND  OTHER  LOANS 


Standard  Form,  ]\^o.  3.t. 


No.  (XXVII)  6. 


To  fiiul  tlie  equated  annual  inoome  to  he  received  over  the  whole 
of  the  unexpired  period  which  is  equivalent  to  the  varying 
amounts  of  income  to  he  received  during  the  first  and 
second  parts  of  such  period  respectively.  Tahle  III. 

Eequired  the  annual  instalment  to  he  set  aside  and  accumulated 
at  3  per  cent,  to  provide  £5165"82  in  13  years. 

(C)     By  Thoman's  Table,  3  per  cent.  Rule  3,  Chapter  XIII. 


Loo".  Amount  of  loan     ... 
add  Log.  (7",  13  years 


51G5-82      3T131393 
8-9732643 


dedmct  Loo^.  E^+IO 


Eequired  annual  instalment.. 


12-6864036 
10-1668839 

2-5195197 

£330-765 


Standard  Form,  No.  3. 


No.  (XXTII)  7. 


To  find  the  amount  which  will  he  in  the  fund  at  the  end  of  the 
first  part  of  the  unexpired  period  of  repayment,  being  the 
accumulation  of  the  amended  annual  increment  during 
that  period.  Table  III. 

Eequired  the  amount  of  an  annual  increment  of  £1077-357  to  be 
added  to  the  sinking  fund  and  accumulated  for  8  years  at 
3  per  cent. 

(B)     By  Table  III,  8  years,  3  per  cent.      Eule  2,  Chapter  YI. 

Log.  Amount  of  £1  per  annum  ...    889234       0-9490159 
r/rW  Locj.  annuitv 1077-357     3-0323597 


Eecjuired  future  amount 


3-9813756 
£9580-22 


APPENDIX 


495 


Standard  Form,  No.  1. 


No.  (XXVII)  8. 


To  find  the  amount  of  loan  wliich  will  be  provided  at  the  end 
of  the  repayment  period,  bein^  the  accumulation  durinf,^ 
the  second  part  of  such  period  of  the  amount  in  the  fund  at 
the  end  of  the  first  part.  Table  I. 

Required  the  amount  of  £9580'220  at  the  end  of  5  years 
accumulated  at  3  per  cent. 

(B)     By  Table  I,  5  years,  3  per  cent.  Eule  2,  Chapter  IV. 


Log.  Amount  of  £1 

add  Log.  present  sum 


Required  future  amount 


11593 
9580-22 


0-0641861 
3-9813756 

40455617 

£1110610 


Standard  Form,  No.  3. 


No.  (XXVII)  9. 


To  find  the  amount  of  loan  which  will  be  provided  by  the 
accumulation  of  the  amended  annual  increment  during  the 
second  part  of  the  unexpired  repayment  period.     Table  III. 

Required  the  amount  of  an  annual  increment  of  £1027-693  to 
be  added  to  the  sinking  fund  and  accumulated  for  5  years 
at  3  per  cent. 


(C)     By  Thoman's  Table,  3  per  cent. 

Log.  Annuity 

q^fZ  Log.  RN^  5  years,  +  10 


Rule  3,  Chapter  VI. 

1027-693      30118636 
10  0641861 


deduct  Loo;,  a^' 


Required  future  amount 


130760497 
9-3391623 

3-7368874 

£5456165 


Index 


INDEX. 

Accounting  methods.     See,  Book-keeping  methods. 

Accounts.     /See  Bank  accounts. 

investment  accounts. 
pro  forma  accounts, 
sinking  fund  accounts, 
suspense  accounts. 

Accounts  of  Local  Authorities,  report  of  Departmental  Committee  (1907),  xii. 

Accounts,  standard  forms  of,  British,  gas  works,  xii. 
electric  lighting,  xii. 
tramways,  xii. 

Accumulating  sinking  fund,  referred  to  in  all  cases  as  the  sinking  fund  unless 
term  non-accumulating  is  used,  126.     Sec  Sinking  fund. 

Accumulation,  rate  of.     See  Rate  of  accumulation. 

Additional  burden  imposed,    on   equation,    on   the    final  years  of  the  equated 
period,  as  regards,  the  annua!  instalment,  410. 
interest  upon  the  loan,  412. 
the  total  annual  loan  charges,  413. 
charts  or  diagrams,  454-7. 
undertaking  may  have  to  incur  outlay  on  renewals,  416. 

Adju.stment,  causes  of,  7,  146. 

Adjustment,  methods  of,  8,  17. 

annual  increment  (balance  of  loan)  method,  8,   152,  260. 

annual  increment  (ratio)  method,  9,   151,  263,  279. 

deductive  method,  224. 

direct  method,  237. 

statement  shewing  full  details  of  each  adjustment  will  be  found  at  the  end 

of  each  chapter,  8. 
summary  of  methods  will  be  found  at  the  head  of  each  chapter,  8. 

Advantage  of  method  by  formula,  159. 

Advantage  of  use  of  logarithms,  3,  21. 

Algebraical  formulpe.     See  Formulae. 

Algebraical  theory  of  indices,  and  its  relation  to  logarithms,  24. 

Alternative  nature  of  the  methods  of  repayment  in  Sec'tion  234  of  the  Public 
Health  Act,  1875,  6,  109,  110. 

Amended  annual  increment.     See  Annual  increment. 

Amended  annual  instalment.     See   Annual  instalment. 

American  experts  to  the  National  Civic  Federation  of  New  York,  1906,  xi. 

American  readers,  to,  ix.  • 

Amount,  as  used  in  published  tables,  must  not  be  confounded  with  a  sum  ot 
money;  a  better  term  would  be  "accumulated  amount"  or  "accumu- 
late," 31. 


500  INDEX 

Amount  due  at  the  end  of  any  number  of  years  :  to  provide  the  same  by  an 
annual  sum  or  annuity  to  be  accumulated  at  compound  interest,  126. 

Amount  in  the  sinking  fund,  deficiency,   154,  171. 
surplus,  186,  199. 
investments,  present,  147. 

Amount  of  an  annuity.     Sue  Amount  of  one  pound  per  annum. 

Amount  of  one  pound,  the,  in  any  number  of  years,  arithmetical  method,  37,  38. 
derivation  of  formulje  :  mathematical,  29,  38. 

Thoman's,  77. 
formula  :  mathematical,  36. 

Thoman's  method,  77. 
Inwood's  table,  No.  1,  36. 
logarithmic  methods  of  calculation,  36. 
rate  per  cent,  per  annum,  to  find,  89. 

ditto,   standard  calculation  form,  89.  ^ 

rules  for  calculations  :  by  formula,  rule  1,  36. 

by  published  tables  of  compound  interest,  rule  2,  36. 

by  Thoman's  method,  rule  3,  37. 
standard  calculation  form,  author's,  No.  1,  40,  41,  88. 
years,  number  of,  to  find,  89. 

ditto,  standard  calculation  form,  89. 

Amount  of  one  pound  per  annum,  in  any  number  of  years,  arithmetical  method, 

54-56. 
derivation  of  formulte  :  mathematical,  55. 

Thoman's,  77. 
formulae  :  mathematical,  50. 

Thoman's  method,  73,  77. 
Inwood's  table.  No.  3,  50. 
logarithmic  methods  of  calculation,  50. 
rate  per  cent,  per  annum,  to  find,  94. 

ditto,  standard  calculation  form,  94. 
rules  for  calculations  :  by  formula,  rule  1,  51. 

by  published  tables  of  compound  interest,  rule  2,  51. 

by  Thoman's  method,  rule  3,  51. 
standard  calculation  form,  author's,  No.  3,  GO,  61,  93. 
yea.rs,  number  of,  to  find,  94. 

ditto,  standard  calculation  form,  94. 

Amount   of   one   pound,    and  of  one  pound   per   annnum   at  end   of  one  year, 
comparison,  33. 

Annual  charges  to  revenue  or  rate,  under  the  annuity  method,  122. 
instalment  method,  113. 
sinking  fund  methods,  the  accumulating  fund,  139. 

the  non-accumulating  fund,  137. 
comparison  under  all  jnethods,  140. 

Annual  incidence  of  taxation.     Si-e  Equation  of  the  period.     See  Incidence  of 
taxation. 

Annual  income  from  inve.stments.     See  Income  from  investments. 


INDEX  501 

Annual  increment,  advantages  of  methods,  149. 
amended  annual  increment,  definition  of,  260. 
balance  of  loan  method,  152,  260. 
definition  of  terms,  151,  175,  260. 
future  annual  increment,  definition  of,  260. 
methods,  8,  262. 

present  annual  increment,  definition  of,  260. 
problems,  259. 
ratio  method,  9,  151,  263. 
variation  in  rate  of  income  included  in  adjustment,  150. 

Annual  increment  (balance  of  loan)  method,  compared  with  application  of  part 
of  fund  in  redemption  of  debt,  153,  175. 

definition  of  terms,  260. 

description  of,  152. 

rates  of  income  and  accumulation  must  be  uniform  during  whole  of  repay- 
ment period,  175. 

summary  of  method,  261. 

under  this  method  amount  in  fund  may  vary  from  proper  calculated 
amount,  205. 

Annual  increment  (ratio)  method,  assumed  that  the  fund  stands  at  the  proper 

calculated  amount  as  shewn  by  the  pro  forma  account,  227. 
definition  of  terms,  260,  281. 
description  of  method,  151,  263. 
principles  of  method,  259,  279,  311. 

rate  of  income  from  investments,  merged  in  annual  increment,  2? 8. 
will  apply  equally  to  an  increase  or  reduction,  282,  299. 
rate  of  accumulation  only  :  summary  of  method,  277. 

rule,  277. 

derivation  of  rule  and  formula,  278. 

formula,  281. 

pro  forma  account,  235. 

method  described,  279. 
period  of  repayment  only  :  summary  of  method,  283. 

rule,  295. 

derivation  of  rule  and  formula,  296. 

formula,  299. 

pro  forma  account,  294. 

method  described,  296. 
rate  of  accumulation  and  period  of  repayment  in  combination  :  summary  of 

method,  300. 

rule,  301. 

derivation  of  rule  and  formula,  304. 

formula,  306,  307. 

pro  forma  account,  321. 

method  described,  303. 

proof  of  method,  307. 

Annual  instalment,  equal,  of  principal  and  interest  combined  :  «n  of  Thoman 
gives  log  values  of,    118. 
annuity  method  of  repayment,  114. 
annuity  which  one  pound  will  purcliase,  67. 


5o:J  INDEX 

Annual  instalment,  equal,  of  principal  and  interest  combined  :  apportionment 

between  capital  and  income,  122,  133. 
compared  with  sinking  fund  instalment,  118. 
formulae,  67. 

hire  purchase  sy.stem.  111. 
method  of  calculating,   115. 
number  of  years,  to  find,  104 

ditto,  standard  calculation  form,    105. 
rate  per  cent.,  to  find,  105. 

ditto,  standard  calculation  form,  105. 
relation  to  sinking  fund  instalment,  118. 
repayment  of  loan  by  the  annuity  method,  120,  122. 
rules,  114,  115. 

standard  calculation  form,  No.  5.  103. 
statement  shewing  final  repayment  of  loan,  122. 
Thoman's  method,  67,  68. 
Annual  instalment  of  the  sinking  fund  :  amended  annual  instalment,  definition 

of,  261. 
apportionment  of,  in  respect  of  loan  borrowed  for  part  only  of  first  year, 

365. 
before  and  after  equation  of  the  period,   15,  418. 
compared  with  equal  annual  instalment  of  principal  and  interest  combined, 

118. 
formulae,  126,  127. 

functions  of  annual  instalment,  417. 
future  annual  instalment,  definition  of,  261. 
method  of  calculating,  131. 
number  of  years,  to  find,  97. 

ditto,   standard  calculation  form,  97. 
present  annual  instalment,  definition  of,  261. 
problems  relating  to  the,   145. 

pro  forma  account  of  the  normal  accimiulation  of  the  fund,  168. 
rate  per  cent.,  to  find,  98. 

ditto,   standard  calculation  form,  98. 
relation  to  equal  annual  instalment  of  principal  and  interest  combined,  118. 
rules,   127. 
standard  calculation  form.  No.  3x,  96. 

Annual  instalment  of  the  sinking  fund,  after  equation  :  spread  equally  over  the 
equated  period  without  any  regard  to  the  life  of  the  asset,  14,  409. 
should  be  spread  over  the  equated  period  in  proportion  to  the  life  of  the 

a.sset,  14,  416. 
when  it  may  be  considered  the  equivalent  of  depreciation,  415. 

Annual  instalment  of  the  sinking  fund,  before  equation  :  spread  over  the  period 

in  proportion  to  the  life  of  the  asset,  14,  378. 
Annual  in.-^talment  of  the  sinking  fund,  fir.st.     See  Fir.^t  annual  instalment. 
Annual  sum,  formula  relating  to,  2,  50. 
Annuities  set  aside  at  end  of  year,  33. 

Annuities,  tables  relating  to  :  amount  of,  50. 
present  value  of,  62. 
purchased  with  one  pound,  67. 


INDEX  503 

Annuities  or  other  periodic  payments,  51. 

cannot  be  considered  a  pure  geometrical  progression,  52. 

Annuity,  amount  of.     Ste  Amount  of  one  pound  per  annum. 

Annuity  method  of  repayment,  114. 

advantages  and  disadvantages,   115. 

annual  incidence  of  taxation,  122. 

annual  charges  to  revenue  or  rate  compared  with  instalment  and  sinking 
fund  methods,  140. 

apportionment    of   annual    instalment    as   between    principal   and   interest, 
122,  133. 

commercial  form,  hire  purchase  system,  111. 

instalment  method  compared,   115,   121. 

loanholder  may  out  of   equal  annual   instalment   of  principal  and   interest 
combined,  provide  a  sinking  fund  for  redemption  of  his  capital,  115, 133. 

relation  between  the  equal  annual  instalment  and  the  sinking  fund  instal- 
ment, 118. 

sinking  fund  methods  compared,  133. 

table  shewing  the  operation  of  the   method   and  the  annual   incidence   of 
taxation,  122. 
Annuity  of  which  one  pound  is  the  present  value.      See  Annuity  which  one 
pound  will  purchase. 

may  be  found  by  Thoman's  factor  an,  118. 

Annuity,  present  value  of.     Sec  Present  value  of  one  pound  per  annum. 

Annuity  which  one  pound  will  purchase,  the;    or  of  which  one  pound  is  the 

present  value  :  annuity  method  of  repayment,  relation  to,  69. 
derivation  of  formulfe  :  mathematical,  69. 

Thoman's,  76. 
equal  annual  instalment  of  principal  and  interest  combined,  relation  to,  69. 
formulae  :  mathematical,  67,  68. 

Thoman's  method,  73,  76. 
Inwood's  table.  No.  v,  67. 
logarithmic  methods  of  calculation,  67,  68. 
rate  per  cent,  per  annum  to  find,  105. 

ditto,  standard  calculation  form,   105. 
rules  for  calculations  :  by  formula,  rule  1,  68. 

by  published  tables  of  compound  interest,  rule  2,  68. 

by  Thoman's  method,  rule  3,  68. 
standard  calculation  form,  author's.  No.  5,  71,  72,  103. 
Thoman's  log  factor  an,  76. 
years,  number  of,  to  find,  104. 

ditto,  standard  calculation  form  104. 

Antilogarithm,  26. 

Appendix,  calculations,  after  Chapter  xv,  459. 

Application  of  sinking   fund,  in  repayment   of  loans  or   redemption   of   stock. 

See  Loans  repaid  out  of  the  sinking  fund. 
Apportionment  :    annual    instalment    or   rent,    method    of  finding   apportioned 
part,  29. 


504  INDEX 

Apportionment  :  as  between  capital  and  income  of  equal  annual  instalment  of 

principal  and  interest  combined,   122,   133. 
effect  of  apportioning  sinking  fund  instalment   during  year  of  borrowing, 

365. 
equal  annual  in.stalment  of  principal  and  interest  combined,  122,  133. 
method  of  finding  any  number  of  days'  proportion  of  an  annual  instalment 

or  of  an  annual  rent,  29. 
of  bank  interest  between  several  sinking  funds,  352. 

ditto,  method  of  avoiding  when  only  one  bank  account  is  kept,  352. 
of  income  from  investments  as  between  several  sinking  funds,  352. 

method  of  avoiding,  etc.,   352. 
of  part  of  instalment  to  be   charged   again.st  year  of  borrowing,  may   be 

advisable  in  case  of  large  loans,   10,  366. 

ditto,  is  generally  ignored  in  case  of  small  loans,  10,  365. 
sinking  fund  instalment,  method  of  finding  apportioned  part,  29. 

Appreciation  of  assets,  381. 

Arithmetical  method  of  finding  :  equated  annual  income,  334. 

error  in  method,   335. 

example,  334. 

pro  forma  account,  341. 
period  of  repayment,  360,  392,  404. 

error  in  method,  397,  405. 

example.?,  392,   404. 

may  be  preferred  on  equation  as  it  extends  period,  393,  402,  439. 
proportionate  part  of  annual  instalment,  368. 

Arithmetical  progression  :  definition  of,  22. 
relation  to  geometrical  progression,  22. 

Assessment  or  local  rate  in  Great  Britain  :   levied   on  annual  and  not  capital 
value,  xiii. 
charged  with  any   deficiency   of   loan  charges   not  provided   by  profits   of 
trading  departments,  xv. 

Asset  :  after  equation  of  period,  redemption  charges  should  be  governed  by  life 

of,  416,  439. 
appreciation  of,  381. 
depreciation  of,   346,   385,  415. 

life  of,  and  its  relation  to  the  redemption  period,  11,  377. 
life  of,  generally  ignored  in  fixing  total  redemption  charges  after  equation 

of  period,  416. 

ditto,  should  govern  total  redemption  charges  after  equation  of  period, 
416. 
obsolescence,  346,  385. 
proceeds  of  sale  of,   186,  189. 
supersession,  346,  385. 
wastage  of.     See  Depreciation. 

Author's  standard  calculation  forms.     Set  Calculation  forms. 

Balance  of  loan  method  of  adjustment,  and  the  annual  increment,  summary  of, 
152,  261. 

Balance  sheet,  standard  forms  of,  xii. 


INDEX  505 

Bank  accounts  :  only  one  account  required  for  the  whole  of  the  sinking  funds 
of   each    department    of  the    local  authority,    351.      See.   Book-keeping 
methods. 

Bank,  gain  by,  on  discount  of  bills,  34. 

Bank  interest,  apportionment  of,  between  several  sinking  funds  where  only  one 
bank  account  kept,  352. 
apportionment  of,  to  avoid,  352. 

Bankrupt  or  insolvent  community,   383. 

Bills,   discount  on  :   difference   between  practical  discount  and  true  or  mathe- 
matical discount,  34. 
published  tables  of  present  values  will  not  apply,  35. 

Book-keeping  methods  :  bank  accounts,  number  of,  351. 
bank  interest,  apportionment  of,  352. 

ditto,  to  avoid,  352. 
commercial  undertakings,  160. 

deficiency  to  be  adjusted  annually  by  revenue  or  rate  account,  352 
equated  loan  interest  reserve  account,  445. 
income  from  investments,  apportionment  of,  352. 

ditto,  to  avoid,  352. 
interest  on  loan  after  equation,  reserve  account,  445. 

pro  forma  account,  447. 
interest  (sinking  fund)  suspense  account,  352. 
investment  accounts,  number  of,   351. 

loans  redeemed  out  of  sinking  fund,  interest  to  be  added  to  the  fund,  129. 
loans  redeemed  out  of  sinking  fund,  should  be  treated  as  investments,  and 

not  debited  to  the  sinking  fund,  351. 
non-accumulating  sinking  fund,  138. 
number  of  sinking  funds  required  :  loans  repayable  at  various  dates,  349. 

objections  to  keeping  only  one  account,  350. 

stock  redeemable  on  one  date,  362. 
pro  forma  accounts,  preparation  of,  5.     See  pro  forma  accounts. 

should  be  kept  in  a  separate  book  and  not  in  the  current  ledger,  5. 
separate  sinking  funds  required,  362. 

ditto,  not  required,  353,  363. 
simplified  on  consolidation  of  loans,   414. 
sinking  fund  interest  suspense  account,  138. 
sinking  funds  should  be  earmarked,  350. 
verification  of  calculations,  by  alternative  method  of  proof,  18. 

Borrowing  :  construction  period,  extended  borrowing  during,  347,  356,  391. 
each  year's  borrowings  treated  as  separate  loan  with  separate  sinking  fund, 

348. 
first  year  of,  and  proportion  of  annual  instalment,  365. 
general  practice,  345. 

in  advance  of  requirements,  results  of,  347. 
may  be  complicated  by  variation  in  life  of  asset,  346. 
relation  between  date  of  borrowing  and  redemption  period,  345. 
loan  relating  to  outlay  of  one  character  only,  borrowed  over  several  years, 
in  one  sum  in  each  year,  each  year's  borrowing  being  repayable 
in  prescribed  period  from  date  of  borrowing,  345. 


5o6  INDEX 

Borrowing  :  example  to  illustrate,  348. 
particulars  of  loan,  349. 

alternative  methods  of  keeping  sinking  funds  :  one  fund  with  increas- 
ing instalment,  349. 
objection  to  method,  350. 

separate  fund  for  each  year's  borrowings,  350. 
each  fund  should  be  earmarked,  with  department,  date  of  sanction  and 

loan,  and  period  allowed,  350. 
bank  account,  only  one  required,  351. 
investment  account,  only  one  required,  351. 
book-keeping  methods,  352. 
loan  relating  to  outlay  of  one  character  only,  borrowed  over  several  years, 
in  one  sum  in  each  year,  repayable  in  one  sum  on  a  specified  date. 
Where  date  of  repayment  is  known  at  the  time  money  borrowed, 
354. 
summary  of  methods,  354. 
how  it  may  arise,  356. 
example  to  illustrate,  357. 
particulars  of  loan,  358. 
one  sinking  fund  only  required,  358. 
increasing  annual  instalments,   358. 
final  repayment  of  loan,  359. 
loan  relating  to  outlay  of  one  character  only,  borrowed  over  several  years, 
in  one  sum  in  each  year,  repayable  in  one  sum  on  a  specified  date, 
fixed  after  fund  has  been  in  operation  a  number  of  years,  359. 
summary  of  methods,  355. 
consolidation  of  existing  loans,  356. 
temporary  instalments  during  construction,  357. 
equation  of  period  of  repayment,  360. 
example  to  illustrate,  360. 
particulars  of  loan,  361. 
sinking  funds  required,  362,  363. 
proof  of  method,  364. 
loan  relating  to  outlay  of  one  character  only,  borrowed  at  various  dates 
in  one  or  more  years,  and  it  is  required  that  the  revenue  or  rate 
account  of  each  year  shall  be  charged  with  a  proportionate  part  of 
the  annual  instalment, 
adjustment  not  required  in  case  of  small  loans,  365,  367. 
may  be  necessary  in  case  of  large  loans,  366. 
method  of  adjustment,  367. 
example  to  illustrate,  367. 

arithmetical  method  of  finding  proportionate  part  of  year,  368. 
proof  of  method,  372. 

compari.son  with  instalments  when  adjustment  not  made,  373. 
effect  of  adjustment,  374. 

Buildings,  outlay  upon,  380. 

Burden.     S'u:  additional   burden. 

See  earlier  years  of  the  equated  period. 
See  final  years  of  the  equated  period. 
Sec  post  equated  period. 


INDEX  507 

Calculations,  are  numbered  to  shew,  in  bold  type,  the  chapter  to  which  they 
relate, 
on  author's  standard  forms  are  given  at  the  end  of  each  chapter, 
after  Chapter  xv  are  given  in  appendix  only,  459. 

should  in  all  cases  be  verified  by  an  alternative  method  of  proof,  5,  17. 
for  periods  other  than  years,  44,   81. 
calculations  other  than  on  standard  forms,  83,  84. 
method  by  step,  183,  338,  429. 

Calculation  forms,  author's  standard,  4,  73. 
advantage  of,  81. 
thi'ee  methods  on  each  form,  82. 
amount  of  one  pound.  No.  1,  40,  41,  88. 

calculations  made  on  form,  nature  of,  87. 

rate  per  cent.,  to  find,  89. 

number  of  years,  to  find,  89. 
amount  of  one  pound  per  annum,  or  annuity.  No.  3,  60,  61,  93. 

calculations  made  on  form,  nature  of,  92. 

rate  per  cent.,  to  find,  94. 

number  of  years,  to  find,  94. 
annuity  method  of  repayment,  No.  5,  103. 

rate  per  cent.,  to  find,  105. 

number  of  years,  to  find,  104. 
annuity  which  one  pound  will  purchase.  No.  5,  103. 

calculations  made  on  form,  nature  of,  102. 

rate  per  cent.,  to  find,  105. 

number  of  years,  to  find,  104. 
present  value  of  one  pound.  No.  2,  91. 

calculations  made  on  form,  nature  of,  90. 

rate  per  cent.,  to  find,  89. 

number  of  years,  to  find,  89. 
present  value  of  one  pound  per  annum,  or  annuity.  No.  4,  65,  66,  99. 

calculations  made  on  form,  nature  of,  98. 

rate  per  cent.,  to  find,  101. 

number  of  years,  to  find,  100. 
rules  for  calculations.  Stu  liules. 
sinking  fund  method  of  repayment,  No.  3x,  96. 

calculations  made  on  form,  nature  of,  95. 

rate  per  cent.,  to  find,  98. 

number  of  years,  to  find,  97. 
special  calculation  forms,  list  of,  83. 

Calculation  of  a  typical  fund,   158. 

Calculations,  rules  for.     Ser  Rules  for  calculations. 

Capital  of  communities,  consists  of  power  to  levy  a  rate,  377  . 

Causes  of  adjustment,  7,  146. 

Causes  of  an  equation  of  the  period,  13,  413. 

Cessation  of  annual  contributions,  221. 

Characteristic,  the  integral  part  of  a  logarithm,  25,  26. 
negative,  26. 

ditto,  to  divide  or  multiply  a  log  with  a,  26. 


5o8  INDEX 

Charges,  annual,  to  revenue  or  rate,  under  instalment  method,   113. 
annuity  method,  122. 
sinking  fund  methods,  accumulating  sinking  fund,   139. 

non-accumulating  sinking  fund,   137. 
comparison  under  all  methods,   140. 

Charts,  or  diagrams,  454-457. 

Clauses,  model,  of  the  Local  Government  Board,  6,   110. 

Commercial    and    financial,    or   private    undertakings  :    annual    in.stalment   not 

always  a  charge  upon  protits,  191. 
auditors  and  position  of  fund,  161. 
book-keeping,  sinking  fund,  methods,   160. 

conditions  as  to  repayment  much  more  elastic  than  local  authorities,  285. 
conversion  of  loan  debt  into  ordinary  share  capital  or  stock,  199,  203. 
deferred  payment  system,  111. 
deficiency  in  the  sinking  fund,  160. 
general  practice  as  to  sinking  funds,  109. 
hire  purchase  system,  111. 
incorporation  of,  xiv. 

investment  of  fund  in  outside  securities,  147,  160. 
limited  liability,  xiv. 
loan  debt  of,  xiv. 

loan  repayment  and  depreciation  kept  distinct,  415. 
methods  of  fixing  annual  instalment  to  sinking  fund  :  by  calculation,  203. 

by  a  fixed  annual  sum,   203. 
methods  of  repayment,  109. 

outside  investment,  if  not  required,  transactions  are  book  entries  only,  160. 
parliamentary  companies,  xiv. 
period  of  repayment,  variation  in,  285. 
proceeds  of  sale  of  assets  paid  into  fund,  compared  with  local  authorities, 

189. 
redemption  by  drawings,    220. 

repayment  of  loans  distinct  from  depreciation,  415. 
reserve  funds  unlimited  in  amount,  385,  415. 
share  capital  or  stock,  methods  of  raising,  xiv. 
sinking  funds  compared  with  those  of  local  authorities,  202. 
sinking  fund  not  always  specifically  invested,  160. 
surplus  in  fund  and  how  corrected,  199. 

surplus  in  fund  arising  on  conversion  of  loan  into  ordinary  share  capital  or 
stock  :  where  original  instalment  found  by  calculation,  204. 

where  original  instalment  a  stated  sum,  210. 
unlimited  liability,  xiv. 
withdrawal  of  part  of  loan  from  operation  of  fund,  203. 

Committee  on  Investigation,  National  Civic  Federation  of  New  York,  ix. 
Companies  or  corporations  in  Great  Britain,  xiii. 

Comparison  of  :  amounts  and  i)resent  values  of  one  pouml  and  of  one  pound  per 
annum  at  end  of  one  year,  33. 
annual  charges  to  revenue  or  rate  inider  all  methods  of  repayment,  140. 
annual  instalments  under  annuity  and  sinking  fund  methods,  118. 


INDEX  509 

Comparison  of  :  arithmetical  and  mathematical  methods  of  finding  the  equated 
date  of  repayment,  392,  397. 

depreciation  and  the  sinking  fund  instalment,  385,  415. 

local  authorities  and  private  undertakings,  conditions  imposed  as  to  repay- 
ment of  loan  debt,   109. 

incidence  of  taxation  before  and  after  the  equation  of  the  period  of  repay- 
ment, 413. 

instalment  method  and  non-accumulating  sinking  fund  method,  135. 

Compound  interest,  applied  to  : 

a  sum  of  money  :  amount  of  one  pound,  36. 

present  value  of  one  pound,  42. 
an  annual  or  other  periodic  payment  :  amount  of  one  pound  per  annum,  50. 

present  value  of  one  pound  per  annum,  62. 
annuity   which  one   pound   will    purchase  or   of  which   one  pound    is   the 

present  value,  67. 
derivation  of  formula  relating  to,  29. 
formula,  relating  to  a  sum  of  money,  2. 

ditto,  an  annual  sum,  2. 
geometrical  progression,  a,  29. 

ditto,   algebraical  formula  for,  30. 
mathematical    formula    derived   from    algebraical   formula    for    geometrical 

progression,  31. 
.symbols,  explanation  of,  30,  84. 
tables  of,  varieties,  3. 
Thoman's  tables,  3. 

Thoman's   method    and   formulae   are   stated   at   the   head    of  the    Chapter 
relating  to  each  Table  and  also  in  Chapter  ix. 
Compound  interest,  published  tables  of.     See  Tables  of  compound  interest. 
Con.'^olidation  of   existing    loans,    additional  burden  on  final  years  of  equated 
period,  410,  412,  413. 
advantages  of,  413. 
amount  in  the  sinking  funds,  13,  361. 
cause  of  an  equation  of  the  period  of  repayment,  11. 
claim  that  it  resulted  in  an  immediate  saving  in  rates,  17,  452. 
effect  on  different  departments  of  local  authority,  11. 
equitable    method    of    adju.sting    the   annual    incidence    of    taxation    after 

consolidation,  409,   418,  436. 
incidence  of  taxation,  annual,  effect  upon,  434,  449. 

ditto,  previous  to  consolidation  should  not  be  varied,  435. 
incidence  of  loan  charges  on  different  departments  of  the  local   authority 

should  not  be  varied  on  consolidation,   11. 
increased  burden  upon  final  years  of  equated  period,  after  consolidation, 

449. 
interest  upon  the  loan  before  and  after  consolidation,  412. 
loanholder,  consent  of,  12. 

method  of  ascertaining  the  equated  date,  389. 
powers  of  Local  Government  Board,   285. 
relief  to  early  years  of  equated  period,  413. 

ditto,  post-equated  period,  413. 
sinking  fund  instalment  before  and  after  consolidation,  410. 
total  annual  loan  charges  before  and  after  con.solidation,  413. 


5IO  INDEX 

Consols,  variation  in  the  rate  of  interest  on,  225,  327. 

Construction  period,  and  its  relation  to  the  sinking  fund  instalment,  347,  356. 
adjustment  of  fund  required  when  works  completed,  356,  391. 
estimated  amounts  only  can  be  set  aside  during  construction,  391. 
matter  complicated  when  works  include  portions  with  varying  periods  of 

continuing    utility,    with    varying    periods    of    repayment,    346.       .See 

Temporary  instalments. 

Continuation  of  annual  instalments,  222. 

Continuing  utility.     ,SV^   Asset.     Life  of   asset. 

Conversion  of  part  of  loan  debt  of  a  private  undertaking  into  share  capital  or 
stock,  and  its  effect  upon  the  sinking  fund,  199. 

Corporations    in   Great   Britain,    include    local  authorities    and    private   under- 
takings, xiii. 

Correct   method  of  di.'itributing  annual    burden  after   equation  of  period,  427, 
436,  449. 

Correction  of  a  deficiency  in  the  sinking  fund  of  a  local  authority  or  private 
undertaking  :    by    additional    annual    instalment    spread    equally    over 
whole  of  unexpired  portion  of  repayment  period,  173. 
by  additional  annual  instalment  spread  over  earlier  part  only  of  unexpired 

portion  of  repayment  period.  179. 
by  an  immediate  payment  into  the  fund,  162. 

Correction  of  a  surplus  in  the  sinking  fund  :  of  a  local  authority,  186. 
of  a  private  undertaking,  199. 

Co.sts  of  obtaining  powers  :  effect  of  equation  of  period  to  extend  repayment  to 
full  term,   15. 
periods  allowed  for  repayment  generally  short,  129. 

Dates  of  borrowing  :  in  one  sum  in  each  year,  over  several  years,  each  yearly 

amount  being  repayable  over  a  similar  period,  345. 
ditto,  the  whole  loan  being  repayable  in  one  sum  on  a  specified  date,  354. 
in  several  sums  in  one  year,  and  it  is  required  to  charge  the  revenue  or 

rate  account  of  the  year  of  borrowing  with  an  apportioned  part  of  one 

year's  annual  instalment,  365. 
may  be  complicated  by  a  variation  in  the  life  of  the  asset,  346. 
apportionment  of  annual  instalment  in  year  of  borrowing,  365. 
relation  between  date  of  borrowing  and  the  redemption  period,  345. 
sinking  fund,  loan  generally  treated  as  being  borrowed  at  end  of  financial 

year  and  full  annual  instalment  set  aside  at  end  of  following  year,  365. 

See  also  Construction   period. 

Date  of  repayment,  10,  285,  345,  354. 

Deductive  method  of  adju.stment,  224,  283. 

Deferred   payment   system,    a   commercial    form    of   the    instalment    method  of 
repayment,  111. 

Deferred  sinking  fund,  391. 


INDEX  511 

Deficiency  in  the  fund  :  causes,  7,  154,  161. 

compared  with  method  of  adjusting  a  surplus,  193,  219. 

depreciation  in  value  of  investments,  161. 

if  small  in  amount  should  be  corrected  annually,  146. 

income  from  investments,  decrease  in,   161. 

methods  of  adjustment,  summary  of,  155. 

methods  will  apply  to  a  surplus,  226. 

methods  of  correcting,  161. 

by  payment  of  deficiency  into  fund,  162. 

by  additional  instalment  spread  over  whole  of  unexpired  period,  162. 
summary  of  method,  171. 
method  described,  173. 
pro  forma  account,  178. 
by   additional   instalment   spread  over   early  years   only   of  unexpired 
period,  162. 
summary  of  method,    172. 
method  described,    179. 
pro  forma  account,   185. 
Stock  Eegulations,  County,  1891,  Art.  11  (2),  161. 

Departments  of  local  authority  :  one  investment  account  and  one  bank  account 
only  required  for  each,  351. 
should  be  kept  distinct  on  consolidation  of  loans,  11. 

Departmental    committee  on   the    accounts    of    local   authorities,    report    dated 

1907,  xii. 
Depreciation  of  assets  :  and  its  relation  to  the  sinking  fund  instalment,  346. 
may  be  considered  as  provided  by  the  annual  instalment  when  the  period 

of  repayment  is  fixed  with  regard  to  the  life  of  the  asset,  385,  415. 
should  be   charged   against  revenue  or   rate  account   when  total    loan  has 
been  repaid  by  means  of  the  sinking  fund,  386. 
Derivation  of   formute  and  rules    (annual  increment  (ratio)    method)  :  rate   of 
accumulation,  variation  in,  277. 
period  of  redemption,  variation  in,  295. 

rate  of  accumulation  and   period  of  redemption  in  combination,   variation 
in,  301. 
Derivation  of  formula  (mathematical  methods)  :  amount  of  one  pound,  36. 
amount  of  one  pound  per  annum,  50. 
annuity  method  of  repayment,  114. 
annuity  which   one  pound    will   purchase   or   of    which  one   pound   is   the 

present  value,  67. 
equal  annual  instalment  of  principal  and  interest  combined,  67. 
present  value  of  one  pound,  42. 
present  value  of  one  pound  per  annum,  62. 
sinking  fund  method  of  repayment,    126. 
Diagrams  or  charts,  shewing  incidence  of  taxation  before  and  after  the  equation 

of  the  period,  454-457. 
Difference  between  the  amounts  of  £1  and  of  £1  per  annum  at  the  end  of  one 

year,  33. 
Direct  method  of  adjustment,  237. 


512  INDEX 

Discount,  mathematical  or  true,  compared  with  practical  discount,  34,  35. 
Discount  on  bills,  compared  with  mathematical  or  true  discount,  34. 
Discount  practical,  published  tables  of  present  values  will  not  apply,  35. 
Division,  by  logarithms,  23,  24. 

Drawing.?,  redemption  of  loans  by,  in  case  of  commercial  and  financial  under- 
takings, 220. 
Duration  of  continuing  utility,  or  the  life  of  the  asset  and  its  relation  to  the 

redemption  period  and  the  incidence  of  taxation.   111,  377,  389. 
main  factor  in  fixing  the  original  period  of  repayment  of  the   loan,   378, 

390,  413. 
Local  Government  Board,  powers  of,  as  to  determining  the  period,  111,  390. 
ignored  generally  when  fixing  the  equated  period  of  repayment  of  loans  in 

respect   of  outlays   of  varying  nature ;     and    also   on  consolidation  of 

existing  loans,  416. 
Earlier  years  of  the  equated    period  :   relief,    on   equation,  annual   instalment 

only,  410,  449. 
interest  on  loan,  unaltered  on  equation,  412,  449. 
total  annual  loan  charges,  413,  449. 
corresponding  burden  on  final  years  of  equated  period,  413. 

Early  provisions  as  to  repayment  of  local  loans,  109,  379. 

Effect  of  the  generally  adopted  method  after  equation  as  regards  :  the  annual 

instalment,  15,  410,  418.     Chart  I,  454-5. 
the  interest  on  the  loan,  16,  412,  436.     Chart  I,  454-5. 
the  total  annual  loan  charges,  16,  413,  449.     Charts  II  and  III,  456-7. 
the  incidence  of  taxation,  15,  16,  409. 
saving  in  annual  rates  claimed  on  equation,  17,  452. 
loans  in  respect  of  outlay  having  a  short  life,  not  repaid  by  the  time  the 

asset  is  worn  out,  420. 
reborrowing  in  respect  of  assets  having  a  short  life,  15,  420. 

Electric  lighting  undertakings  :  standard  forms  of  account,  xii. 

wide  divergence  in  charge  by  different  municipalities,  383. 

See  also  Revenue  earning  undertakings. 
Enquiry  into   the  municipal  and   private   ownership    and   operation   of   public 

utilities  in  Great  Britain ;   by  the  National  Civic  Federation  of  New  York 

(1906),  ix. 
Equal  annual  instalment  of  principal  and  interest  combined  :  may  be  found  by 
Thoman's  factor  an,  89. 

See  Annuity  which  one  pound  will  purchase,  67. 

Equated  annual  income  from  investments.  330. 
arithmetical  method  of  finding,  334. 
error  in  method,  335. 
pro  forma  account,  341. 

Equated  date,  method  of  ascertaining,  adopted  by  Local  Government  Board, 
402,  403. 
generally  fixed  by  method  known  as  the  equation  of  payments,  392,  404. 
error  in  method,  397,  405. 
true  or  mathematical  method  described,  394,   399,  404. 


INDEX  513 

Equated  loan  interest,  reserve  account,  after  equation  of  period,  445. 
pro  forma  account,  447. 

Equated  period  :  necessity  to  fix,  how  arising,  13,  389. 

arithmetical  method  or  the  equation  of  payments,  14,  334,  392,  404. 
arithmetical  method  gives  longer  period,  402. 
true  or  mathematical  method,  394,  404. 

summary  of  method,  399. 
on  the  consolidation  of  loans,  389. 
for  repayment  of  loans,  raised  by  the  issue  of  stock,  in  respect  of  outlay 

upon  works  having  varying  periods  of  continuing  utility,  390. 
relief  to  early  years  on  equation  of  period  in  respect  of  the  :  annual  instal- 
ment, 15,  410,  449. 
interest  on  the  loan,  412,  449. 
total  annual  loan  charges,  413,  449. 
charts  or  diagrams,  454—457. 
further  borrowing  powers  after  equation,  420. 

additional  burden  imposed  after  equation  of  period  upon  the  final  years  of 
the  equated  period  in  respect  of  the  :  annual  instalment,   15,  410, 
449. 
interest  on  the  loan,  16,  412,  449. 
total  annual  loan  charges,  413,  449. 
charts  or  diagrams,  454-7. 
number  of  years  in  period   depends  upon  amount   of   loan  and  respective 

repayment  periods,  405. 
See  Earlier  years  of  equated  period. 

Equation,  method  of  :  as  generally  adopted,  14,  410. 
effect  of,   15,  413. 
true  or  mathematical  method,  394,  399,  404. 

Equation  of  payments,  the  arithmetical  method  in  general  use  for  finding  the 
equated  date,  14,  334,  392,  404. 

Equation  of  the  incidence  of  taxation  :  before  equation  the  annual  instalment 
and  interest  on  the  loan  are  borne  by  future  years  in  proportion  to  the 
life  of  the  asset,  438. 
after  equation,  life  of  asset  generally  ignored,  434. 

the  unequal  incidence  of  taxation  if  the  annual  instalment  and  interest  on 
the  loan  are  spread  equally  over  the  equated  period  without  any 
regard  to  the  life  of  the  asset,  409. 
remedy  as  to  the  annual  instalment,  15,  431. 
ditto,  interest  on  the  loan,  16,  443. 
method  of  continuing  the  original  annual  instalments  during  the  equated 
period,  and  spreading  the  supplementary  annual  instalment,  represent- 
ing  the   relief   to   the   post-equated   period,   equally  over   the  equated 
period,   15,   16,   424. 
method  of  spreading  the  burden  over  the  equated  period  strictly  in  propor- 
tion to  the  life  of  the  asset  before  equation,  16. 
the  annual  instalment,  431. 
the  intere-st  on  the  loan,  443. 
the  total  annual  loan  charges,  449. 
claim  to  reduce  rates  on  consolidation,  17,  452. 


514  INDEX 

Equation  of  the  incidence  of  taxation  :  cannot  be  made  by  reducing  each  period 
in  proportion  to  the  reduction  in  the  final  period,  419. 
difference   between  financial   obligation  to   repay  loan,   and   the   charge  to 

revenue  or  rate  based  upon  life  of  asset,  419. 
charts,  454—457. 

Equation  of  the  period  of  repayment  :  and  the  life  of  the  asset,  11,  389. 

and  the  incidence  of  taxation,  15,  409. 

on  the  consolidation  of  existing  loans,  391. 

claim  to  reduce  rates  on  consolidation,  17,  452. 

in  respect  of  a  loan  authorised  for  outlays  having  varying  periods  of  con- 
tinuing utility  and  consequent  repayment,  309. 

and  effect  in  case  of  renewals,  416. 

postponed  repayment  of  loans  in  respect  of  outlay  having  a  life  shorter 
than  the  repayment  period,  15,  420. 

causes  giving  rise  to  equation,  13,  413. 

life  of  asset  enters  into  calculation  on  equation  but  effect  lost  by  equal 
annual  instalment,  424. 

a  financial  operation  only,  409,  415. 

effect  of  equation,  409,  411,  434. 

Evolution  by  logarithms,  24. 
Expenses  of  sinking  fund,   129. 

Final  years  of  the  equated  period.  See  Additional  burden,  on  equation,  and 
Equation  of  the  incidence  of  taxation. 

Final  repayment  of  the  loan,  8.  .S'ec  Statements  which  will  be  found  at  the 
end  of  each  chapter  dealing  with  an  adjustment  in  a  smking  fund,  ^'ee 
also  the  several  pro  forma  accovmts. 

Finance  Act,  1888,  225,  327. 

Financial  details  of  undertakings  in  Great  Britain  examined  by  the  Committee 
of  Investigation  of  the  National  Civic  Federation  of  New  York  in  1906,  x. 

Financial  nature  of  the  equation  of  the  period,  415,  453. 

Financial  undertakings.     See  Commercial  undertakings. 

First  annual  instalment  to  sinking  fund  :  method  of  ascertaining  the  propor- 
tionate part  of  the  annual  instalment  in  respect  of  moneys  borrowed 
for  part  of  a  year  to  be  charged  against  the  rate  or  revenue  account  of 
the  year  of  borrowing,  365. 
usually  set  aside  at  the  end  of  the  first  complete  year  and  no  charge  made 
against  the  year  in  wliich  the  loan  is  borrowed,  365. 

Forms  of  accounts.     See  Accounts. 

Forms  for  calculations,  author's  standard,  these  forms  are  fully  described  and 
explained  in  Chapter  x. 

FormuliE  (annual  increment  ratio  method)  :  rate  of  accumulation,  variation  in, 

281. 
period  of  redemption,  variation  in,  299. 
rate  of  accumulation  and  period  of  redemption  in   combination,   variation 

in,  307. 


INDEX  515 

Formula  :  advantages  of  method  by,  3,  158. 
annuities  or  other  periodic  payments,  50. 
geometrical  progression,  2,   30. 
compound  interest,   2,   31. 

simple  interest,  28. 
differ  from  those  in  actuarial  works,  3. 

Formulae  (mathematical  methods)  :  amount  of  one  pound,  36. 
amount  of  one  pound  per  annum,  50. 
annuity  method  of  repayment,  67,   114. 
annuity   which    one   pound    will    purchase   or   of  which   one   pound   is  the 

present  value,   67,  68. 
equal  annual  instalment,  of  principal  and  interest  combined,  67,  68. 
present  value  of  one  pound,  42. 
present  value  of  one  pound  per  annum,  62. 
sinking  fund  method  of  repayment,  127. 
derivation  of.     See   derivation   of  formulfe. 

Future  amount,  present  value  of.     See  Present  value  of  one  pound. 

Future  annual  income  from  investments,  definition  of,  260.     -S'ee  Income  from 
investments. 

Future  annual  increment,  definition  of,  260. 

Future  annual  instalment,  definition  of,  261. 

Future  generations,  liability  to  provide  further  utilities,  378,  384. 
safeguards  to,  14,  378,  384,  410,  412,  413. 

Future  rate  of  accumulation,  definition  of,  260. 

Gas  works,  profits  from,  applied  in  aid  of  rates,  383. 

wide  divergence  of  charges  by  different  municipalities,  383. 

standard  forms  of  account,  xii. 

See  also  under  Revenue  earning  undertakings. 

Geometrical  progression  :  algebraical  formula  for,  2,  30. 
basis  of  formula  relating  to  compound  interest,  2,  30. 
definition  of,  22,  30. 

derivation  of  formula  relating  to  compound  interest,  30,  31. 
example  of,  22. 
relation  to  logarithms,  22. 

Goschen  (Finance  Act,   1888),  225,  327. 

Hire  purchase  system,  a  commercial  form  of  the  annuity  method  of  repayment, 
111. 

Incidence  of   loan   charges  on  different  departments  of   a    local  authority,    on 

consolidation  of  loans,  11. 
Incidence  of  taxation.     See  Ratepayer. 
Incidence   of  taxation,    equation  of   the.      See  Equation  of   the   incidence   of 

taxation. 


5i6  INDEX 

Incidence,  annual,   of  taxation,   under  various  methods  of  repayment  :   instal- 
ment method,  113. 
annuity  method,  122. 
sinking  fund  methods  :  the  accumulating  fund,  139. 

the  non-accumulating  fund,  137. 

comparison  under  all  methods,   140. 

Income  from  investments  :  consols,  variation  in  the  rate  of  interest  on,  225,  327. 
future  annual  income,  definition  of,   149,  260. 
present  annual  income,  definition  of,  260. 
non-accumulating  sinking  fund,  135. 

rate  of,  variation  in  :  uniform  during  whole  of  period,  236. 
not,  ditto,  322. 
See  also  Variations. 

Increment,  annual,   148.     See  Annual   increment. 

Indices,  algebraical  theory  of,  24. 

Injustice  to   ratepayers,    of    final    years   of    equated   period.       ^ee    Additional 
burden,  ratepayer. 

Instalment,  equal  annual,  of  principal  and  interest  combined,  may  be  found  by 
Thoman's  factor  an,  118.     See  Annual  instalment,  equal,  of  principal  and 
interest  combined. 

Instalment  method  of  repayment,  109. 

annual  incidence  of  taxation,  under  this  method,   113. 

annual  charges,  to  revenue  or  rate,  compared  with  :  annuity  method,  115, 
121. 

sinking  fund  methods,  128. 
commercial  form,  the  deferred  payment  system.  111. 
decreasing  annual  charge  to  revenue  or  rate,  112. 
insurance  companies,  111. 
loanholder,  frequent  reinvestment  by,   112. 
Public  Works  Loan  Commissioners,  111. 
no  accumulating  sinking  fund,  113. 

relation  to  the  non-accumulating  sinking  fund  method,  135. 
statement  shewing  the  operation  of  the  method  and  the  annual  incidence 

of  taxation,  113. 

Instalment,  annual,  to  sinking  fund,  problems,   145. 

Interest  :  compound,  a  geometrical  progression,  29. 
simple,  an  arithmetical  progression,   28. 

Interest,  bank.     See  Bank  interest. 

Interest  from  investments.     See  Income  from  investments. 

Interest  of  one  pound,  for  one  year,  45,  52. 

basis  of  finding  tlie  formula  relating  to  the  accumulation  of  an  annual  or 

other  periodic  sum,  52. 
Symbol  r,  32,  85. 
values  and  corresponding  logs  for  49  rates  per  cent.,  from  ^  per  cent,   to 

7  per  cent..  Table  V,  A,  Chapter  V,  48,  49. 

Interest   suspense  account,    sinking  fund,  method  of  keeping,   relating  to  one 
bank  account,  and  one  investment  account  for  several  sinking  funds,  352. 


INDEX  517 

Interest  upon  the  loan  :  after  repayment  of  loan  out  of  the  accumulatmg  sink- 

ijig  fund  to  be  added  to  the  fund,  in  whole  or  in  part,  129. 
annuity  method,   apportionment   of    equal    animal    instalment  as    between 

principal  and  interest,  122. 
annuity  method,    decreasmg    annual   amount    of    interest   included  in   the 

equal  annual  instalment  repaid  to  the  lender,  115. 
annuity  method,  loanholder  may   provide   a  sinkmg   fund  to  equalise  his 

annual  interest,  133. 
charge  to  revenue  or  rate  account,  under  all  methods  of  repayment,  140. 

before  and  after  consolidation,  412,  436. 

before  and  after  equation  of  period,  412,  436. 

effect  of  the  equation  of  the  period,  416. 
consent  of  loanholder,  on  consolidation,  12. 
instalment    method,    decreasing   annual    amount    of   interest    paid    to    the 

lender,  112,  113. 
need  not  be  added  to  non-accumulating  fund  after  repayment  of  loans,  137. 
the  incidence  of  taxation,  416,  436. 

reserve  account,  to  equalise  the  incidence  of  taxation  after  the  equation  of 
the  period  of  repayment,  pro  forma  account,  447. 

book-keepmg  methods,  445. 
sinking    fund    method,    constant    annual   amount   of    interest   paid   to   the 

lender,  129. 
variation  in  the  mcidence  of  taxation  after  the  equation  of  the  period  of 

repayment,  436. 

Introduction,  1. 

Investment  accounts  :  only  one  investment  account  required  for  the  whole  of 
the  sinking  funds  of  each  department  of  a  local  authority,  351. 
should  be  raised  in  the  ledger  for  all  loans  repaid  or  redeemed  out  of  the 
sinking  fund,   351.     iSee  also  Book-keeping  methods. 

Investment  of  the  sinking  fund  in  loans  of  the  same  local  authority,  130. 

Investments  representing  the  sinking  fund  and  the  income  arising  therefrom. 

See  Income  from  investments. 
Investments,    value  of,    an  important  factor   in   any   adjustment   of  a   sinkuag 

fund,    147. 
Investments,  rate  of  income  from.     -S'ee  Kate  of  income  from  investments. 
Involution,  by  logarithms,  24. 

Inwood's  tables,  of  compound  interest,  etc.,  new  edition  by  Schooling,  3. 
Irredeemable  stock,  12. 
Irregular  contributions  to  the  sinking  fund  in  earlier  years.     See  Construction 

period.     See  Temporary  instalments  to  the  sinking  fund. 

Land,  appreciation  in  value,  382. 
acquisition  by  way  of  lease,  380. 
period  allowed  for  repayment  of  loan,  380. 

Law  relating  to  the  repayment  of  the  loan  debt  of  local  authorities,  is  not 
included  in  this  book  except  so  far  as  it  relates  to  the  actual  methods  of 
repayment,  1. 


5i8  INDEX 

Leasehold  projjerties,  outlay  upon,  by  local  authorities,  380. 

Life,  or  duration  of  continuing  utility,  of  the  asset  created  out  of  the  loan,  IL 
and  its  relation  to  the  redemption  period,  377. 
after  the  equation  of  the  period  of  repayment,  -116,  439. 
See  also  Assets. 

Limitation  of  the  period   of   repayment,   in  the  case  of  works  of  almost  per- 
manent utility,   14. 

Limited  Liability  Acts  (Great  Britain),  xiv. 

Loan   charges,    annual,    charged   against  profits  of  trading   departments  :    any 
deficiency  made  good  out  of  annual  rates  or  assessments,  xv. 

Loan  debt  of  local  authorities,  how  raised  :  mortgages  for  short  terms,  414. 
stock,  390. 
security  for,  383. 

Loan  debt  of  local  authorities,  how  repayable  :  by  instalment  method,  109. 
by  annuity  method,   114. 
by  sinking  fund  methods,  126. 
at  end  of  short  periods,  mortgages,  414. 
on  a  specified  date  (stocks),  390. 
out  of  the  sulking  fund,  129. 
See  reborrowing. 

Loan  debt  of  private  undertakings,   how  repayable.      See  Loan  debt  of  local 
authorities,  commercial  undertakings. 

Loan,  final  repayment  of.     See  Statements  shewing,  etc.,  pro  forma  accounts. 

Loanholder  :  and  the  equation  of  the  period  of  rejaayment,  414. 

annuity  method,  difficulty  in  ascertaining  the  amount  of  principal  included 

in  each  annual  in.stalment,   115. 
ditto,  effect  of  annual  repayments,  115. 
may  provide  a  sinkmg  fund  for  the  redemption  of  his  capital  out  of  the 

equal  annual  mstalment  of  principal  and  interest  combined,  115. 
consent  required  on  consolidation  of  existing  loans,  12,  383. 
fluctuating  nature  of  investment  under  the  instalment  and  annuity  methods, 

112,  115. 
interests  considered  on  equation  of  period,  409. 
instalment  method,  effect  of  repayment  by,  112. 
commercial  and   financial   undertakings,    conversion   of  loan   into  ordinary 

share  capital  or  stock,  203,  210,  213. 
security  for  local  loans,  383. 

preferential  nature  of  repayment  out  of  sinking  fund,  409. 
reinvestment,    frequent,    of    caj)ital,    under    the    instalment    and    annuity 

methods,  112,  115. 
sinking  fund  method,  a  permanent  investment,   134. 
option  to  convert  into  ordinary  share  capital  or  stock,  203. 

Loans,  consolidation  of.     See  Consolidation  of  loans. 

Loans,  conversion  of,  of  a  private  undertaking,  into  ordinary  share  capital  or 
stock,  199. 

Loans  fund,  definition,  129. 


INDEX 


519 


Loans  of  local  authorities  :  how  raised,  and  how  repayable.     Sac  l^oan  debt  of 
local  authorities. 

Loans  in  respect  of  outlay  having  a  short  life,  not  repaid  after  equation  by  the 
time  the  asset  is  worn  out,  15,  420,  435.     Sac  also  Reborrowing. 

Loans  repaid  and  stock  redeemed  out  of  the  sinking  fund  :  application  of  whole 

or  part  of  fund  authorised  by  Public  Health  Act,   1875,  Sec.  234  (5), 

129. 
ditto,  non-accumulating  sinking  fund,    137. 
interest  to  be  added  to  the  fund  to  make  up  the  deficiency  in  the  fund 

which  would  be  caused  by  such  application  of  the  fund,  130. 
interest  to  be  added  to  the  fund  to  be  equivalent  to  the  interest  which 

would  liave  been  produced  by  the  amount  so  applied,  130. 
need  not  be  added  in  the  case  of  the  non-accumulating  fund,  137. 
should  be  treated  as  an  investment  of  the  fund,  and  not  debited  to  the 

sinking  fund  account,  28,   130. 
Stock  purchased  at  a  premium ;    par  value  only  may  be  taken  out  of  the 

fund,  the  premium  to  be  charged  against  the  current  year's  revenue 

or  rate  account  or  against  a  reserve  created  for  the  purpose,  i30. 

Local  authorities,  accounts  of,  report  of  the  Departmental  Committee,  1907,  xii. 

Local  authorities,  principles  as  to  borrowing  and  repayment  laid  down  in  the 
Public  Health  Act,  1875,   110. 

Local  authority,  definition  of  term,  xiii. 

Local  Government  Board  :  auditors,  and  the  pro  forma  account,  5. 
County  Stock  Regulations,  1891,  deficiency  in  fund,  161. 
Local  Government  Act,  1888,  285. 

method  of  equating  the  period  of  repayment,  actual  example,  402. 
model  clauses  (1893),  See. 

not  represented  on  parliamentary  committees,  379. 
powers  as  to  period  of  repayment.  111. 
powers  as  to  permanency  of  works,  110. 
powers,  on  consolidation  of  loans,  285. 
practice  as  to  fixing  periods  of  repayment,  1. 
pro  forma  accounts,  copies  should  be  sent  to  Board,  in  relation  to  all  loans 

coming  under  the  supervision  of  the  Board,  5. 
Public  Health  Act,  1875,  powers  under,  110. 
Public  Health  Act,  Amendment  Act,  1890,  powers  under,  285. 
sale  of  assets,  proceeds  of,  190. 
supervision  by  Board  avoided  by  proceeding  by  Special  Act,  379. 

Local  rate,   or  assessment  in  Great  Britain,   levied  on  the  annual  value,   and 
not  on  the  capital  value,  xiii,  17. 

Logarithms,  Chapter  II.  :  advantages  of,  3,  21. 
algebraical  theory  of  indices,  24. 
antilog,  to  find,  26. 

arithmetical  progression,  definition  of  22. 
Briggean  or  decimal  logs,  22. 
characteristic,  definition  of,  25. 

of  numbers  greater  or  less  than  unity,  rules,  26. 
common  or  decimal  logarithms,  22. 


520  INDEX 

Logarithms  :  definition  of,  23. 

divide  one  log  by  another,  to,  27,  398. 
division  by,  23,  24. 
earliest  published  tables  of,  21. 
evolution  by,  24. 

fractional  part  or  mantissa,  always  positive,  24. 
geometrical  progression,  algebraical  formula,  2. 
ditto,  definition  of,  22. 
ditto,  relation  to  compound  interest,  2. 
history  of,  21. 

interest  of  one  pound  for  one  year  for  49  rates  per  cent,   from  ^  per  cent. 
7  per  cent,  logs  of.  Table  V,  A,  48,  49. 

integral  part  or  characteristic  may  be  either  positive  or  negative,  26 
involution  by  logs,  24. 

mantissa,  definition  of,  24. 
always  positive,  24. 

multiplication  by  logs,  23,  24. 

negative  characteristic,  rule  to  multiply  or  divide  a  log  with,  26. 

ratios  for  49  rates  per  cent,  from  ^  per  cent,  to  7  per  cent.  Table,  Va,  48,  49. 

relation  between  arithmetical  and  geometrical  progression,  22,  23. 

tables  of  common  logarithms,  21. 

Thoman's  logs  of  an  increased  by  10  to  avoid  negative  characteristic,  75,  79. 

to  find,  of  even  powers  of  10,  23. 
Logarithmic  formulae  and  rules.     iSce  Formulae,  rules. 

Logarithmic  tables,  of  compound  interest  and  annuities,  Thoman's,  73. 
of  numbers  from  1  to  108,000  to  seven  places  of  decimals,  26. 

Loss  on  sale  of  investments  representing  the  sinking  fund,  161. 
Mantissa,  the  fractional  part  of  a  logarithm,  24. 
Manufacturing  plants,  outlay  upon,  384. 
Markets,  outlay  upon,  381. 

increasing  value  of  site,   381. 
Mathematical  formulae  and  rules.     Sec   Formulte. 
Mathematical  method  of  equation  of  period,  394,  399,  404.     .'Sec  Equated  date, 

methods  of  obtaining, 
^lathematical  or  true  discount,  compared  with  practical  discount,  34. 
Mathematical  method  by  formula,  indispensable  if  rate  per  cent,  not  given  in 

any  published  table,  4,  5,    159. 
Mathematical  principles,  2,  19. 
Mathematical  tables,  varieties  of,  3.     Schooling,  3.     Inwood's,  3.     Thoman's,  73. 

Sprague's,  3. 
Method  by  step,  of  finding  the  amount  of  an  annuity  for  a  term  of  years  accu- 
mulated for  a  further  period,  183,  338,  429. 
Methods  of  adjustment  described  :  annual  increment  methods,  8. 

annual  increment  (balance  of  loan)  method,  8,  152,  260. 

ditto,  (ratio),  9,  151,  263,  279. 

deductive  method,  224. 

direct  method,  237. 

statement  shewing  full  details  of  each  adjustment  will  be  found  at  the  end 
of  the  Chapter,  8. 

summary  of  methods  will  be  found  at  the  head  of  each  Chapter. 


INDEX  521 

Methods  of  ascertaining  equated  date.     ISee  Equated  date. 

Methods  of  book-keeping.     6'ee  Book-keeping. 

Methods  of  calculation,  82. 

Methods  of  repayment  of  loan  debt,  107.     Sat  Loan  debt  of  local  authorities. 

Model  clauses  of  the  Local  Government  Board,  6,  110. 

Mortgages,  for  short  terms,  414. 

Multiplication,   by  logarithms,  23. 

Municipal   and   private   ownership  and  operation  of   public   utilities  in  Great 

Britain ;    enquiry  by  the  National  Civic  Federation  of  New  York,   1906 ; 

report,  1907,  ix. 
Municipal  trading,  ix. 

National  Civic  Federation  of  New  York,  ix. 

enquiry  into  the  municipal  and  private  ownership  and  operation  of  public 

utilities  in  Great  Britain,  1906,  ix. 
financial  details  of  undertakings  examined,  x. 
committee  of  investigation,  list  of  members,  ix. 
experts  engaged  on  the  enquiry,  xi. 
report,  dated  1907,  xi. 
Non-accumulating  sinking  fund  :  accumulation  of  the  fund  under  this  method, 
137. 
advantages  and  disadvantages,  136,  137,  138. 
annual  incidence  of  loan  charges,   136. 
application  of  part  of  fund  in  repayment  of  debt,  137. 
book-keeping  methods,   138. 

compared  with  instalment  and  annuity  methods,   136. 
description  of  the  method,  135. 
first   introduced   in    the  model    clauses    of   the    Local   Government   Board, 

1893,  6,  110. 
income  from  investments,  how  treated,   135. 
objects  of,  135. 

relation  to  the  instalment  method,  135. 

statement  shewing  the  operation  of  the  method  and  the  annual  incidence 
of  taxation,  137. 
Number  of  bank  accounts  required.   -S'ee  Bank  accounts.    Book-keeping  methods. 
Number  of  investment  accounts  required.      See  Investment  accounts.      Book- 
keeping methods. 
Number  of  sinking  funds  required  :  in  respect  of  loan  borrowed  over  a  series 
of  years  and  repayable  at  different  dates  over  a  similar  period,  objec- 
tions to  keeping  one  sinking  fund  only,  350. 
in  respect  of  a  loan  borrowed  over  a  series  of  years,  redeemable  in  one  sum 
on  a  certain  specified  date,  353. 
Number  of  years  to  find.     .See  Calculation  forms. 

may  be  found  exactly  by  formula  in  the  following  cases  :  amount  of  one 
pound,  89. 
present  value  of  one  pound,  89. 
amount  of  one  pound  per  annum,  94. 
sinking  fund  instalment,  97. 


522  INDEX 

Number  of  years  to  find  :   may  be   found   approximately   by  reference   to  the 
published    tables    in    the   following    cases  :    present   value    of    one 
pound  per  annum,   100. 
annuity  one  pound  will  purchase,  104. 
method  fully  described,  397. 
in  equated  period  depends  upon  two  factors,  405. 

Objections  to  keeping  one  sinking  fund  for  loans  repayable  at  different  dates, 
350. 

Obsolescence  of  assets,  346.     See  Assets. 

Occupation  of  lands  by  local  authorities  by  lease  rather  than  purchase,  380. 

One  pound,  amount  of.     See  Amount  of  one  pound. 

One  pound,  interest  of,  for  one  year.     See  Interest  of  one  pound,  etc. 

One  pound,  present  value  of.     See  Present  value  of  one  pound. 

One  pound  per  annum,  amount  of.     See  Amount  of  one  pound  per  annum. 

One  pound  per  annum,  present  value  of.     See  Present  value  of  one  pound  per 

annum. 
Outlay  having  short  period,  and  effect  of  equation,  15,  420,  425.  435. 

Outlay  having  varying  periods  of  repayment  included  in  one  sanction,  346,  390. 

Outlay  upon  manufacturing  plants,   384. 

Outlay  upon  renewals  during  later  years  of  equated  period,  416. 

Ownership  and  operation  of  public  utilities  in  Great  Britain,  ix. 

Parks  and  open  spaces,  outlay  upon,  380. 

Parliament,  variation  in  conditions  allowed,  379. 
investment  of  the  sinking  fund,  130. 
final  approval  required  for  all  loans,  379. 
committees  of,  379. 
policy  as  regards  repayment  periods,  382. 

Parliamentary  Companies  in  Great  Britain,   xiv. 

Past  rate  of  accumulation,  definition  of,    260.     See  Rate  of  accumulation. 

Payments,  equation  of.     See  Equation  of  payments. 

Periodic  payments.     See  Annuities. 

Periods  shorter  than  one  year,  method  of  calculation,  44. 

Period   of    construction   and   relation   to    sinking   fund.     See    Construction 
period. 

Period  of  continuing  utility.     See   Asset,   life  of  asset. 

Period  of  repayment,  equation  of.     See  Equation  of  the  period. 


INDEX 

Period  of  redemption  or  repayment  : 

commercial   and  linanciai    undertakings,    285. 
determined  with  relation  to  the  life  of  the  asset,  377. 
equation  of  the  period  a  financial  operation,  415,  453. 
may  be  varied  on  consolidation  of  loans,   285. 
rarely  altered  in  case  of  individual  loans  of  local  authority,  285. 
relation  to  the  date  of  borrowing,   345. 
sub.stituted  repayment  period,   definition,  261. 
unexpired    repayment    period,    definition,    261. 
variable  nature  of  conditions  now  existing,  379. 

variation  in  the  period  without  any  variation  in  the  rate  of  accumu- 
lation :   summary  of  method,   283. 

rule,  295. 

derivation  of  rule  and  formula,   296. 

formula,    299. 

pro  forma  account,    294. 

method   de.scribed,    286,    296. 
variation  in  the  period  with  a  variation  in  the  rate  of  accumulation  : 

summary   of   method,    300. 

rule,   301. 

derivation  of  rule  and  formula,  304. 

formula,  306,  307. 

pro  forma   account,    321. 

method  described,  303. 

proof  of  method,    307. 

Permanence  of  works.     See  Asset,  life  of.  Local  Government  Board. 

Permanent  utility,   works  of  almost,   limit  imposed  as    to  period  of  repay- 
ment,   14,    380,    382. 

Perpetual  debt  of  local  authorities,  379. 

Plants,   manufacturing,   outlay  on,  384. 

Post  equated  period,   relief  to,   should  be  spread  over  the  equated  period 

in  proportion  to  the  life  of  the  asset,  416,  449. 
under  present  method  on  equation  is  spread  equally  over  the  equated 

period,   409,   449. 
relieved  of  all  charges  in  resiject  of  annual  instalment  and  interest  on 

loan,   413,   449. 
charts  or  diagrams,  454-7. 

Practical  discount  compared  with  true  or  mathematical  discount,  34,  35. 
published  tables  of  present  values  will  not  apply,  35. 

Preface,   v. 

Preface,  to  American  readers,  ix. 

Preferential  nature  of  repayment  by  means  of  sinking  fund,  409. 

Preliminary  stages  in  sanction  of  a  loan,  care  bestowed  to  find  the  proper 

period   to  be  allowed  for   repayment,    14,   415. 
Premium,  redemption  of  stock,   at  a,   221. 

Premium  paid  on  purchase  of  stock  out  of  sinking  finid,   12. 
may  not  be  taken  out  of  sinking  fund,   130. 


523 


524  INDEX 

Present  annual  income  from  investments.     Sec   Income   from  investments. 

definition  of,   260. 
Present  annual  increment,   definition  of.     Sec   Annual  increment. 
Present  annual  instalment,   definition  of.     .S'er   Annual  instalment. 
Present  investments,    value   of,    147. 

Present  sum,  to  find  the  amount  of.     See  Amount  of  one  pound. 
Present  value  of  one  pound,  the,  due  at  the  end  of  any  number  of  years,  42. 

derivation  of  formulae,   mathematical,   43. 
Thoman's,    44. 

formulae,   mathematical,    42. 
Thoman's  method,   73,   77. 

Inwood's   table,    No.    2,   42. 

logarithmic  method  of  calculation,  42. 

rate  per  cent,  per  annum,  to  find.  89. 

ditto,   .standard   calculation  form,    89. 

rules  for  calculations,   by  formula,   rule   1,   43. 

by  published  tables  of  compound  interest,  rule  2,  43. 
by  Thoman's  method,  rule  3,  43. 

standard  calculation  form,  author's.  No.  2,  46,  47,  91. 

years,   number   of,   to   find,    89. 

ditto,  standard  calculation  form,  89. 

Present  value  of  one  pound  per  annum,  the,  for  any  number  of  years,  62. 
derivation  of  formulae,  mathematical,  63. 

Thoman's,    78. 
formulae,  mathematical,  62. 

Thoman's  method,  73,  78. 
Inwood's  table.  No.   4,  62. 
logarithmic  methods  of  calculation,   62. 
rate  per  cent  per  annum,  to  find,   101. 
ditto,  standard  calculation  form,   101. 
rules  for  calculations,  by  formula,  rule  1,  63. 

by  published  tables  of  compound  interest,  rule  2,  63. 

by  Thoman's  method,  rule  3,  63. 
standard  calculation  form,  author's,  No.  4,  65,  66,  99. 
years,   number   of,   to  find,    100. 
ditto,   standard   calculation   form,    100. 

Principles   governing   the   borrowing   and    repayment   of   the   loan   debt   of 

local  authorities,   xi,   110. 
Private  ownership  and  operation  of  public  utilities,  ix. 
Private   undertakings.     See    Commercial    undertakings. 

Problems,  sinking  fund,  the  annual  instalment,   Sec.   III. 
the  amount  in  the  fund,   145. 

a  deficiency,  154,  171. 

a  surplus,    186,    199. 
the  rate  of  income   from  investments,    236,    322. 
the  rate   of   accumulation,   223. 
the  rates  of  income  and   accumulation,   247. 


INDEX 


525 


Problems,  sinking  fund,  the  annual  increment.   Section  IV. 

the  rate  of  accumulation,    259,   277. 

the  redemption  period,   283,   295. 

the  rate  of  accumulation  and  the  period  in  combination,  300. 
Proceeds  of  sale  of  assets,  forming  part  of  the  security  for  the  loan,  7,  189. 

applied  in  repayment  of  loan,    189. 

may  be  paid  into  sinking  fund  and  accelerate  final  repayment,   190. 

may   be  applied  in  reduction  of  future  annual   instalment  over  whole 
or  part  of  unexpired  period,   190. 
Profits  of  trading  departments  applied  in  aid  of  rate,  381,  383. 
Profit  and   loss  account,   standard   forms  of,   xii. 
Profit  on  sale  of  investments  representing  the  sinking  fund,  188. 
Pro  forma  accounts,   should  be  prepared  in  all  cases  to  shew  final  repay- 
ment,  5,   17,   145. 

should  be  kept  in  separate  book,  5. 

amended  account  should   be  prepared   after  any  future  adju.stment  of 
the  fund,  5. 

copy  should  be  sent  to  the  Local  Government  Board,  5. 

Local    Government   Board   Auditors,    145. 
Pro  forma  accounts  : — 

1.  Normal  accumulation  of  the  fund,  168. 

2.  Correction  of  a  deficiency  by  additional   annual  instalment  spread 

over  whole  of  unexpired  period,   178. 

3.  Correction  of  a  deficiency  by  additional  annual  instalment  spread 

over  early  years  only  of  unexpired  period,  185. 

4.  Correction    of   a   surplus   by   a   reduced    annual    instalment    spread 

over  whole  of  unexpired  period,  after  payment  into  the  fund  of 
proceeds  of  sale  of  assets,   198. 

5.  Correction  of  a  surplus  by  a  reduced  annual  instalment  spread  over 

whole  of  unexpired  period,  after  withdrawal  of  part  of  loan 
from  operation  of  fund,   209. 

6.  Correction   of   a    surplus   by   a   reduced    annual    instalment   spread 

over  whole  of  unexpired  period,  after  withdrawal  of  part  of 
loan  from  operation  of  fund.   219. 

7.  Correction  of  a  deficiency  by  an  increased  annual  instalment  spread 

over  whole  of  unexpired  period,  consequent  upon  a  reduction  in 
the  rate  of  accumulation,  without  any  variation  in  the  rate  of 
income  from  investments,   235. 

8.  Correction  of  a  deficiency  by  an  increased  annual  instalment  spread 

over  whole  of  unexpired  period  consequent  upon  a  reduction  in 
the  rate  of  income  from  investments  without  any  variation  in 
the  rate  of  accumulation,  246. 

9.  Correction  of  a  deficiency  by  an  increased  annual  instalment  spread 

over  whole  of  unexpired  period  consequent  upon  a  variation  in 
the  rates  of  income  from  investments  and  accumulation,   255. 

10.  Correction  of  a  deficiency  by  an  increased  annual  instalment  spread 

over  whole  of  substituted  period  consequent  upon  a  variation 
in  the  period  of  repayment,  294. 

11.  Correction  of  a  deficiency  by  an  increased  annual  instalment  spread 

over  whole  of  .substituted  period  consequent  upon  a  variation  in 
the  rate  of  accumulation  and  in  the  period  of  repayment,  321. 


526  INDEX 

Pro  forma  accounts  : — 

12.  Correction  of  a  deficiency  by  an  increased  annual  instalment  spread 

over  whole  of  unexpired  period  consequent  upon  a  variation  in 
the  rate  of  income  from  investments  to  occur  at  a  known  future 
date,  339. 

13.  Correction  of  a  deficiency  by  an  increased  annual  instalment  spread 

over  whole  of  unexpired  period  consequent  upon  a  variation  in 
the  rate  of  income  from  investments  to  occur  at  a  known  future 
date,  based  upon  the  equated  annual  income,  3-tl. 
'  Equated  loan  interest.  Reserve  account  :  shewing  the  method  of 
charging  the  equal  annual  amounts  of  interest  payable  to  the 
loanholders,  after  equation,  to  the  revenue  or  rate  accounts  of 
each  year,  in  proportion  to  the  life  of  the  a.sset,  447. 

Progressions,   arithmetical,   28. 

geometrical,   29,    30. 
Proportionate  part,  of  annual  instalment,   10. 

Public   Health  Act,    1875,   alternative  methods  of  repayment,    110. 
expenses  of  sinking  fund,   129. 
interest  on  part  of  sinking  fund  applied  in  redemption  of  debt  to  be 

paid    into   the   fund,    129. 
limit  to  total  amount  to  be  borrowed.   111. 
limitation  of  period  for  purposes  of  the  Act,  111. 
Local  Government  Board  to  decide  for  what  purposes  money  may  be 

borrowed,  389. 
methods  of  repayment,   110. 

permanency  of  works  must  be  taken  into  account.  111,  390. 
power  to  apply  whole  or  part  of  sinking  fund  in  redemption  of  debt, 

129. 
power  to  borrow  limited,   110. 
provisions  as  to  borrowing  and  repayment  of  loans  of  local  authorities, 

6,  109. 
regulations  as  to  exercise  of  borrowing  powers,   109. 

Public    Health  Acts   Amendment   Act,    1890,    285,    390. 

Public  utilities,  early  Acts  of  Parliament  and  repayment  obligations,  379. 
manufacturing  plants,   384. 

many  loans  outstanding  without   any  obligation  as  to  repayment,  379. 
markets,    382. 

National   Civic   Federation   of   New   York,    enquiry,   ix. 
outlay  by  local  authorities,  a  national  question,  379. 
outlay  upon  works  of  permanent  utility  .should  not  be  charged  against 

present  generation,  arguments  for  and  against,  380. 
provided   by   pledging   credit    of   community,    377. 
regard  for  future  requirements,  378,  381. 

repayment  of  outlay  spread  over  period  equivalent  to  life  of  asset,  378. 
safeguard  to  future  generations,   14,  378,   384,  410.  412,  413. 
sewage    disposal,    381. 

should  be  paid  for  in  proportion  to  annual  benefit  received,  377. 
tramways,   383. 

want  of  uniformity  of  practice  in  loan  conditions  now  existing,  379. 
waterworks,   382. 


INDEX  527 

Public  Works  Loan  Commissioners,  and  life  of  asset,  390. 
and   Local   Government   Board,    390. 
instalment   method.    111. 

Published  tables  of  compound  interest.     Sec  Tables  of  compound  interest. 

Rate  of  accumulation,   general   considerations,   225. 

should  be  estimated  on  the  low  side,   in  the  case  of  loans  with   long 

repayment  periods,    148,  326. 
may  be  estimated  slightly  lower  to  allow  for  a  reduction  in  the  future 

rate  of  income   from   investments,    326. 
future   rate   of    accumulation,    definition,    260. 
past   rate   of   accumulation,    definition,    260. 

variation  in,  without  any  variation  in  the  rate  of  income  from  invest- 
ments, summary  of  methods  of  adjustment,  223. 

method  described,  230. 

pro  forma  account,   235. 

annual  increment   (ratio)   method,   277. 
variation  in,   and  also  in  the  rate  of  income  from  investments,   sum- 
mary of  methods  of  adjustment,  247. 

methods  described,  248. 

pro  forma  account,  255. 
variation  in,   and   also  in  period   of   repayment,    summary   of   methods 
of  adjustment,  300. 

methods  described,  303. 

pro   forma  account,    321. 

annual  increment   (ratio)  method,   300. 

Eate  of  income  from  investments  : 

equated  annual   income,   330. 
arithmetical    method    of    finding,    334. 
error   in  method,    335. 
•    pro  forma  account,   341. 
variation    in,    without    any    variation    in    the    rate    of    accumulation, 
summary  of  methods  of  adjustment,  236. 
methods   described,    237,    239. 
pro   forma   account,    246. 
variation  in,  and  also  in  the  rate  of  accumulation,  summary  of  methods 
of   adjustment,   247. 
methods  described,  248. 
pro  forma   account,    255. 
not  uniform  during  unexpired  repayment  period,  322,   327. 
variation    known    at    time    of    making    the    adjustment,    summary    of 
method,    323. 
method  described,  328. 
pro  forma  account,   339. 
variation  anticipated,  but  uncertain  as  to  time  and  amount,  summary 
of  method,   325. 
method  described,  332. 
future  equated  annual  income,  330. 
pro  forma  account,   341. 


528  INDEX 

Ratepayer,   accumulating   sinking   fund  method,    effect   upon   annual   rates, 
134. 
additional  burden  on  equation,  410,  412,  413. 
annual  contribution  towards  benefits  received,  377,  409. 
annuity  method,   effect  upon  annual  rates,    115,   134. 
depreciation  should  be  charged  against  future  generation?,   386. 
immediate  relief  on  equation  of  period,  409. 
injustice  to  future  generations  after  equation,  15,  410,  412,  413. 
instalment  method,  effect  upon  annual  rates,  112.  134. 
interests  on  equation  of  period,  11,   12,  409. 

non-accumulating  sinking  fund,   effect  upon  annual   rates,    136. 
profits  in  aid  of  rate,  381,  383. 

protection  of  future  generations,  14,  378,  384,  410,  412,  413. 
provision  in  advance  for  future  generations,  378. 
relief  to  post  equated  period,  409,  413,  449. 
renewals  during  final  years  of  equated  period,  416. 
repayment  for  assets  of  permanent  utility,  380. 
want   of  interest   in  municipal   matters,    13,    383. 

Rate  per  cent,   general   considerations  as  to,   225. 

Rate  per  cent,   to  find.     See   Calculation   forms. 

may  be  found  exactly  by  formula  in  the  following  cases  : 

amount  of  one  pound,  89. 

present  value  of  one  pound,   89. 
may  be  found  approximately  by  reference  to  the  published  tables  in 
the  following  cases  : 

amount  of  one  pound  per  annum,  94. 

sinking  fund   instalment,   98. 

present  value  of  one  pound  per  annum,   101. 

annuity  one   pound   will    purchase,    104. 

Rate  per  cent.,  variations  in,  rate  of  accumulation  only,  223. 
rate  of  income  from  investments  only,    236. 
rate  of  accumulation  and  income  in  combination,  247. 
statement   showing  original    and   amended   conditions  in   all   variations 
considered,    265,    267. 

Ratio,    definition    of,    in    algebraical    formula    relating    to    a    geometrical 

progression,  30,   45,   52. 
in  mathematical  formula  relating  to  compound  interest,  31. 
values  and  corresponding  logs,  for  49  rates  per  cent,  from  \  per  cent. 

to  7  per  cent..  Table  V,  A,  48,  49. 

Ratio  method  of  adjustment.     The  annual   increment.     See   Annual   incre- 
ment   (ratio)    method. 

Ready  reckoner  tables  of  compound  interest,  etc.,  3. 

Reborrowing,  in  respect  of  assets  having  a  short  life  which  are  worn  out 
before  the  equated  date,   15,  420,   425,  435. 

Redemption  of  .stock  at  a  premium,  221. 
Redemption  of   loan   in  part,  221. 


INDEX 


529 


Redemption  by  drawings,  in  case  of  commercial  and  financial  under- 
takings,   220. 

Redemption   fund,    definition   of,    129. 

Redemption  of  stock  by  purchase  out  of  the  sinking  fund.  Sec  Loans, 
etc.,   repaid   out   of  the   sinking  fund. 

Redemption  period.     Sec  Period  of  redemption  or  repayment. 

Relation  between  amount  of  one  pound  and  of  one  pound  per  annum,  53. 

Relation  between  life  of  asset  and  period  of  repayment,   14,  377. 

Relief  to  early  years  of  equated  period  as  regards  the  : 
annual    instalment,    410. 
interest  on  the  loan,  412. 
total  annual  loan  charges,  413. 
interest  on  the  loan  is  the  same  as  before  equation,  412. 
relief  is  entirely  in  respect  of  the  annual  in!5taiment,  and  is  at  expense 
of  later  years  of  equated  period,  411,  449. 

Relief  to  rates  on  consolidation  of  loans,  instance  given  in  evidence  before 
a  Parliamentary   Committee,    17,   452. 

Renewals,  effect  upon  later  years  of  equated  period,  416. 

Repairs  and  renewals  fund,   385. 

Repayment,    date   of,    9,    10,    345. 

Repayment  of  debt  and  its  relation  to  depreciation.  See  Depreciation  of 
assets. 

Repayment  of  loans  out  of  moneys  in  the  sinking  fund.  See  Loans  repaid 
out  of  the   sinking   fund. 

Repayment,  final,  of  the  loan.  See  Statements,  which  will  be  found  at  the 
end  of  each  chapter  dealing  with  an  adjustment  in  a  sinking  fund. 
See  also  the  several  pro  forma  accounts. 

Repayment  period.     Sec   Period  of  redemption  or  repayment. 

Repayment,  period  of,  equation  of  the.  See  Equation  of  the  period  of 
repayment. 

Repayment  of  loans  by  local  authorities  (1902),  report  upon,  402. 

Reports,   Departmental   Committee  on  the  Accounts  of  Local   Authorities, 
1907,   xii. 

National  Civic  Federation  of  New  York  on  the  municipal  and  private 

ownership  and  operation  of  public  utilities  in  Great  Britain,  ix. 
Repayment  of  Loans  by  Local  Authorities  (1902),  402. 

Reserve  account  for  interest  on  loans  after  equation,  447. 

Revenue  account,   standard  forms  of,  xii. 

Revenue  earning  undertakings  :  appreciation  in  value,  384 
compared   with   .spending  departments,   415. 

depreciation,   covered  by  sinking  fund  when  repayment  period  within 
life  of   asset,    385, 

A  J 


530  INDEX 

Revenue  earning  undertakings:  depreciation  should  be  charged  when  original 
loan  repaid  out  of  sinking  fund,  385. 
land,    outlay   upon,    384. 
obsolete  plant,   385. 
outlay   on   manufacturing  plants,    384. 
outlay  upon  repairs,   384. 
profits  applied  in  aid  of  rate,  383,  415. 
renewals    fund,    385. 
standard   forms   of   accounts,    xii. 
wastage  of  asset,  384,  385. 
wide  divergence  of  charges  by  different  municipalities,   383. 

Rules  for  calculations  (annual  increment   (ratio)  method)  •. 
rate  of  accumulation,   variation  in,   277. 
period  of  redemption,  variation  in,   295. 

rate  of  accumulation  and  period  of  redemption  in  combination,  vana 
tion  in,   301. 

Rules  for  calculations  (mathematical)  : 
amount  of  one  pound,  36. 
amount  of  one  pound  per  annum,   51. 
annuity  method   of   repayment,    68,    114. 
annuity   which    one   pound   will    purchase,    or   of   which   one   pound   is 

the  present  value,   68. 
equal  annual  ini=talment  of  principal  and  interest  combined.  68,   114. 
present  value  of  one  pound,  42. 
present  value  of  one  pound  per  annum,   63. 
sinking   fund   method   of   repayment,    126. 
Thoman's  methods,  are  given  at  the  head  of  each  chapter. 

Sale  of  assets,     ^ee  Proceeds  of  sale  of  assets. 

Sale  of  investments  representing  the  sinking  fund,  loss  on  sale,  161. 
profit    on    sale,    188. 

Sanction  of  a  loan,  preliminary  stages  in  the ;  care  bestowed  to  find  the 
proper  period  to  be  allowed  for  repayment,   14,  415. 

Sanction  for  works  comprising  various  classes  of  outlay  may  specify 
amount  and  period  in  respect  of  each  class,  or  only  total  amount  with 
an  equated  period,  346,  390. 

Saving  in  annual  rates  on  consolidation,   17,  452. 

Schooling,  William,  F.R.A.S.,  author  of  the  new  edition  of  Inwood's 
Tables   (1899),   3. 

Security  for  municipal  loans,  a  mixed  fund  of  capital  and  revenue,  383. 

Separate  sinking  funds.     See   Number  of  sinking   funds. 

should    be    kept    in    all    cases    where    loans    are    repayable    at    various 
dates,  350,  362. 

Share  capital  or  stock  of  private  undertakings,  methods  of  raising,  xiv. 
preferences  and   priorities,   xiv. 
conversion  of  loan  debt  into,   199. 

Short-term   mortgages,   414. 


INDEX  531 

Simple   interest,    arithmetical    progression,    28. 
formulae  and   rules,    28. 

rule   to   find    number   of    days'    proportion   of    an    instalment   or   other 
annual  sum,  29. 

Sinking  fund  : 

accounts,   number   of,    10,   350,    353. 
accumulation,    rate   of,.     See    Rate   of    accumulation, 
accumulating  sinking  fund,   128. 
adjustment,    methods    of.     See    Adjustments, 
amount  in  the  fund,   deficiency,   154,    171. 

surplus,   186,   199. 
annual  charges  to  revenue  or  rate,   137,   139. 
annual   incidence  of  taxation,    139. 
annual    income   from   investments.     Sec    Income, 
annual  increment,   149,   175,   239,   262. 
balance  of  loan  method,  152,  260. 
ratio  method,    151,   263,   279. 
annual    instalment,    method    of    calculating,    131. 
before  and  after  equation,   15,  418. 
first,  365. 
annuity   method,    comparison,    133. 
application  of,  in  repayment  of  debt,  129. 
apportionment   of   one   year's   instalment,    365. 
balance  of  loan  method  of  adjustment,  152,  260. 
bank  accounts,   351. 
bank  interest,   352. 
book-keeping  methods,  see. 
borrowing,   dates  of,   see. 
calculation  of  a  typical  sinking  fund,    158. 
commercial    undertakings,    see. 

comparison  with  annuity  and  instalment  methods,   133. 
consolidation   of   loans,    sec. 
construction,    period    of,    347,    391. 
correction  of  a  deficiency  in  the  fund,  162,  173,  179. 
correction  of  a  surplus  in  the  fund,    186,   199. 
dates   of   borrowing,    see. 
deductive  method  of  adjustment,   224,   283. 
deficiency  in  the  fund,  7,   154,   162,  173,  179. 
depreciation,  relation  to,  346,  385,  415. 
derivation  of  formulae,   131. 
direct  method  of  adjustment,   237. 
equation  of  the  incidence  of  taxation,  409. 
equation  of  the  period  of  repayment,  389. 
expenses  of,  129. 

final  repayment  of  the  loan,   139,   168. 
financial    undertakings,    see     commercial, 
first  annual  instalment,    365 
formulae,  127. 

incidence  of  taxation,   377,   409. 
income  from  investments,   see. 
instalment,    annual,   sec   annual   instalment. 


53i  INDEX 

Sinking  fund   (continued)  : 

interest  suspense  account,   352. 

investment  accounts,  351. 

investments  in  loans  of  same  local  authority,    130. 

investments  of  the   fund,    130,    147,    351. 

investments,  rate  of  income  from,  see  income. 

investments,    sale    of,    188. 

investments,    value   of,    147. 

irregular  contributions  to  the  fund,  346,  356,  367,  391. 

loan,   final  repayment  of,   139,    168. 

Loans  Fund,    129. 

loans  repaid  out  of  the  sinking  fund,  129. 

loss  on  sale  of  investment  of  the,  161. 

method  of  repayment,   126. 

methods  of  adjustment,   see  adjustments. 

normal  accumulation  of  the  fund,  pro  forma  account,   168. 

non-accumulating   sinking   fund,    6,    135. 

ditto,   compared  with  instalment  method,    135. 

number  of  bank  accounts,  351. 

number  of  investment  accounts,   351. 

number   of   sinking   funds   required,    349. 

number  of  years,  to  find,   97. 

period    of    construction,    347,    391. 

period  of  repayment,  283,  295,  300,  345. 

period  of  repayment,  equation  of  the,  see. 

position  of,  to  ascertain,   160. 

premium  paid  on  redemption  of  stock,  130. 

present   investments,    147. 

private  undertakings,   ^ee   commercial. 

problems  relating  to  the  sinking  fund,   145. 

proceeds  of  sale  of  assets,  7,   189. 

profit  on  sale  of  investments,   188. 

pro   forma   account,    normal   accumulation,    168. 

Public  Health  Act,  1875,  see. 

rate  of  accumulation,   see. 

rate  of  income  from  investments,  see  rate  of  income. 

rate  per  cent.,  to  find,   98. 

ratio  method  of  adjustment,  sec  annual  increment  (ratio)  method. 

Redemption  Fund,   129. 

redemption  of  debt,   out  of  sinking  fund,   129. 

redemption  period,   283,    295,   300,   345. 

repayment,  final,  of  the  loan,   139,   168. 

rules  for  calculating  annual  instalment,   127. 

sale  of  assets,    7,    189. 

sale  of  investments,    188. 

separate  funds  for  each  year's  borrowings,  348,  350. 

statements  shewing  the  final  repayment  of  the  loan,   139,  168. 

Stock  redeemed  out  of  the  sinking  fund,  see  loans  repaid,  etc. 

summary  of  methods,   see   methods. 

surplus  in  the  .sinking  fund,   186,    199. 

suspense  account  for  bank  interest  and  income  from  investments,  352, 


inde:x 


53^ 


Sinking  fund   (continued)  : 

temporary  instalments  to  the  sinking  fund,   347,  391. 
typical  sinking  fund,   158. 
variations  in  the  sinking  fund,  see. 
years,   number  of,   to  find,   97. 

Sinking   fund    method    of   repayment,    annual    charges   to   revenue   or   rate 
compared   with   the   instalment   method,    133. 
the    annuity    method,    133. 
table  shewing  the  operation  of  the  method  and  the  annual  incidence 
of    taxation,     139. 

Sinking  fund  problems,   the  annual  instalment,    145. 
the    annual    increment,    259. 
the  date  of  borrowing  and  the  redemption  period,  345. 

Spending  departments  compared  with  revenue  earning  undertakings,  415. 

Sprague's   tables   of   compound   interest,    3. 

Standard   calculation   forms,    author's.     See    Calculation   forms. 

Standard   forms   of   account,   xii. 

Statement   shewing  the  final  repayment   of  tlie   loan   will   be   found   after 
each  adjustment.     Sea  pro   forma  account. 

Statement  shewing  the  methods  of  adjustment  will  be  found  at  the  end 
of   each   chapter. 

Step,   method  by,   of  finding  the  amount  of  annuity  for  a  term  of  years, 
accumulated  for  a  further  period,  183,  338,  429. 

Stock  or  .share  capital  of  private  undertakings  in  Great  Britain,  xiv. 

Stock  redeemed  out  of  the  sinking  fund.     Sec  Loans  redeemed,  etc. 

Stock  regulations,  cessation  of  annual  contributions,  222. 
a  deficiency  in  the  fund,   161. 
necessity  to  equate  period  of  repayment,   390. 

Stocks  issued  by  local   authorities,   the  principal  cause  of  the  equation  of 
the  period  of  repayment,    13,   390. 

Stocks  of  local   authorities  generally,     .b'ec   under  the   various  headings  of 
"Loans,"   "Loan  Debt,"  etc. 

Subject   matter   of  book,    1. 

Substituted   repayment   period,   definition,   261. 

Summary  of  methods  will  be  found  at  the  head  of  the  chapter  dealing  with 
each  adjustment, 
annual  increment  (balance  of  loan)  method,  8,  152,  260. 
annual  increment  (ratio)  method,  9,   151,  263,  279. 
deductive  method,   224 
direct  method,  237. 

Supersession  of  assets,  346. 


534 


INDEX 


Surplus  in  the  fund,   causes  of,   7,    188. 

compared  with  methods  of  adjusting  a  deficiency,  193,  219 
excessive  past  accimiulation  of  fund,  188,  189. 
methods  of  adjustment,   summary  of,    186. 
methods  will  apply  to  a  deficiency,  226. 
of  a  local  authority  and  how  applied,  189,  190. 
payment  into  fund  of  proceeds  of  sale  of  assets,  189. 
summary    of    method,    187. 
method    described,    189. 
realised  profit  on  sale  of  an  investment,  188. 
summary  of  method,   187. 
method  described,   189. 
withdrawal  of  part  of  loan  from  operation  of  fund,    188,   199. 
summary  of  methods  of  adjustment,   199. 
original    annual    instalment    found    by    calculation  : 
summary  of  method.    199. 
method   described,   204. 
original  annual  instalment,  a  stated  sum  : 
summary  of  method,   200. 
method  described,  210. 

Suspense  or  reserve  account  for  interest  on  loans  after  equation,  447. 

Suspense  account,  sinking  fund  interest ;  to  avoid  apportionment,  where 
only  one  bank  account  and  one  investment  account  are  kept  in  respect 
of  several  sinking  funds,  352. 

Symbols   used  in  the  various  formulae,   3. 
explanation  of,  30,  84,  85. 
derivation  of,  30. 

formulae  relating  to  compound  interest  and  annuities,  31. 
geometrical  progression,  30. 


Tables  of  compound  interest,  etc.,  amount  of  one  pound.  Table  I,  36. 
amount  of  one  pound  per  annum.  Table  III,  50. 
annuity  method  of  repayment,  Table  V,  67,  114. 
annuity   which   one   pound    will   purchase,    or   of   which   one   pound   is 

the  present  value,  Table  V,  67. 
equal  annual  instalment  of  principal  and  interest  combined,  67,  114. 
interest   of    one    pound    for   one   year    (?)    from   ^   to   7    per   cent,    and 

logarithms,    48. 
Inwood's  Tables,  3,  33. 
present  value  of  one  pound.  Table  II,  42. 
present  value  of  one  pound  per  annum.  Table  IV,  62. 
ratios,  from  ;^  to  7  per  cent,  and  logs.,  48,  49. 
Schooling,   3. 

Sprague's  Tables   (American),   3. 
Thoman's  Logarithmic  Tables,   3,   73. 
varietie.-  of,  3. 


INDEX  535 

Taxation,   annual  incidence  of,   Tables  shewing,    under  : 
the  instalment  method,   113. 
the  annuity  method,    122. 
the  accumulating  sinking  fund  method,   ^^9. 
the  above  methods  compared,  140. 
the  non-accumulating  sinking  fund  method,   137. 
after  equation.     See   equated  period,   etc. 

Taxation,  equation  of  the  incidence  of.      -S'ee  Equation  of  the  incidence  of 
taxation. 

Temporary    sinking    fund    instalments,    356,    367,    391.     See    Construction 
period. 

Thoman,   Fedor.     Logarithmic  tables  of  compound  interest  and  annuities, 
3,   73. 

Thoman 's  formulae,   methods  and  rules,   advantages  of,   5,   75. 
amount   of  one  pound,    36,    73,    77. 
amount  of  one  pound  per  annum,  50,  73,  77. 

an    log.  of,  stated  by  the  addition  of  10  to  the  characteristic,  75,  78,  79. 
an    represents    the    annuity    which    one    pound    will    purchase,    or    the 

equal  annual  instalment  of  principal  and  interest  combined,  76. 
annuity   method,    67,    114. 

annuity  one  pound  will   purchase,   67,   73,   78. 
equal    annual   instalment   of   principal    and   interest,    67,    114. 
formulae  compared   with  formulae  relating  to  compound  interest,   73. 
present  value  of  one  pound,   42,   73,  77. 
present  value  of  one  pound  per  annum,  62,  73,  78. 
rules  are  stated  both  generally  and  in  logarithmic  form  at  the  head 

of   each   chapter, 
sinking   fund   method,    127. 
symbols  used  by  Thoman,   75. 
tables  are  contained  in  Inwood's  tables, 
tables  give  log.   values  only,   79. 

Town  Halls,  outlay  upon,  380. 

Trading    undertakings.     See    Revenue    earning    undertakings. 

Tramways,  profits  of  undertaking,   383. 

renewals   fund,    385. 

standard  forms  of  account,  xii. 

See   also   Revenue    earning    undertakings. 
True  or  mathematical  method  of  tinding  equated  period  of  repayment,  394, 

399,  404. 

Typical   sinking  fund,   8,   158. 

Unexpired  repayment  period,  261. 
Utility  of  asset.     See  Asset,   life  of  asset. 

Value  of  investments,   147. 


53^  INDEX 

Variations  in,  accumulation,  rate  of,  without  any  variation  in  the  rate  of 
income  from  investments,   223,   277. 
with  a  variation  in  the  rate  of  income  from  investments,  247. 
with  a  variation  in  the  period  of  repayment,   300. 
consols,  rate  of  interest  upon,  225,  327. 
income    from   investments,    rate   of  : 

without  any  variation  in  the  rate  of  accumulation  or  the  period 
of    repayment,    236. 
where  the  future  variation  in  the  rate  of  income  from  invest- 
ments is  definite  both  as  to  date  and  amount,  322,  328. 
where    the    future    variation    is    anticipated    but    is    uncertain 
both  as  to  date  and  amount,  322,   334. 
with  a  variation  in  the  rate  of  accumulation,  247. 
period  of  repayment  : 

without  a  variation  in  the  rate  of  accumulation,  283,  295. 
with  a  variation  in  the  rate  of  accumulation,  300. 
separate  effect  of  the   variation  in   the  : 
rate   of   accumulation,    317. 
period  of  repayment,  316. 

Verification  of  calculations  by  alternative  method  of  proof,   17 

Wastage   of    asset.     Sec    depreciation. 

Waterworks,  381. 

argumenc  against  extended  period  of  repayment,  382. 

argument  in  support  of  extended  period  of  repayment,  382. 

importance  of  outlay  beyond  present  requirements,  381. 

large   amount   invested   in   land,    382. 

large  cost  of  water  areas,   381. 

permanent   character   of   works,    382. 

See   also  under  Revenue  earning   undertakings. 

Works,  permanence  of.     See  asset,   life  of  asset. 
Years,  number  of,  to  find.     See  number  of  years. 


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